On the Collapsing of Manifolds with Local Rewinding Reifenberg Control

Shaosai Huang

Working paper · October 2026

On the Collapsing of Manifolds with Local Rewinding Reifenberg Control

Shaosai Huang

∗ Working paper. Comments welcome.

Kspectra Research Inc., Toronto, Canada

Abstract

Let $(M^n,g_0)$ be a complete Riemannian manifold with $\Ric _{g_0}\ge -(n-1)g_0$ whose local universal covers are uniformly $\delta$ -Reifenberg-close to $\R ^n$ below a fixed radius $\rho$ . We show that for every $\alpha \in (0,1)$ there are $\delta (n,\alpha )>0$ and $T(n,\rho ,\alpha )>0$ such that, if $\delta \le \delta (n,\alpha )$ , a complete Ricci flow $g(t)$ with $g(0)=g_0$ exists on $[0,T]$ and satisfies $|\Rm _{g(t)}|\le \alpha /t$ for $0<t\le T$ . On closed manifolds this is a uniform lower bound for the lifetime of the Ricci flow; on noncompact manifolds, where the initial curvature may be unbounded, it is an existence theorem, and the flow is unique among complete Ricci flows from $g_0$ with a curvature bound of the form $A/t$ . The key step is a localized comparison of two Ricci flows through the Ricci–harmonic map heat flow, following Lee, which transfers a Type-I curvature bound from a complete comparison flow to the incomplete flow lifted to a local cover. Consequently, below a dimensional Reifenberg threshold, collapsing closed manifolds admit singular nilpotent fibrations over their Gromov–Hausdorff limits. Conditional on Nifa’s construction, the Reifenberg hypothesis cannot be weakened to noncollapsing of the local covers: products of shrinking circles with Nifa’s four-manifolds have $\Ric \ge 0$ and uniformly noncollapsed local covers, but admit no continuous Gromov–Hausdorff approximations with small error onto their limit.

Contents

1Introduction
2Preliminaries
3The Local Improvement and the First-Time Argument
4A Complete Comparison Flow
5Localized Comparison
6Completion of the Bootstrap
7Complete Noncompact Initial Manifolds
8Singular Nilpotent Fibrations
9A Counterexample under Rewinding Volume Control
10Discussion
1. 10.1Between the Two Thresholds
2. 10.2Neighbouring Cases
ALocal Ricci–Harmonic Comparison Estimates
References
How to cite

Keywords. Ricci flow; collapsing; Ricci curvature bounded below; local rewinding; Reifenberg property; pseudolocality; singular nilpotent fibration; Gromov–Hausdorff convergence.

2020 Mathematics Subject Classification. Primary 53E20; secondary 53C21, 53C23.

1Introduction

The model is collapse under a two-sided sectional-curvature bound. If closed $n$ -manifolds with $|\sec |\le 1$ collapse to a compact metric space $X$ , Fukaya and Cheeger–Fukaya–Gromov [Fuk87; Fuk89; CFG92] construct $O(n)$ -equivariant nilmanifold bundles $\hat f_i:\Fr (M_i)\to Y_i$ on the frame bundles, and passing to $O(n)$ -orbit spaces gives continuous maps $f_i:M_i\to X=Y_i/O(n)$ with infranilmanifold fibres. Following [Ron26b, Section 1.3], a singular nilpotent fibration is a quotient map arising in this way; the full diagram, with its metric and differential estimates, is recalled in Theorem 8.1.

We ask how much of this survives under a Ricci lower bound alone. A metric ball may then itself be collapsed, so the local hypotheses are imposed on its Riemannian universal cover. We compare two such conditions.

Definition 1.1 (Local Rewinding Conditions). Let $(M^n,g)$ be a Riemannian manifold, $x\in M$ and $\rho >0$ , and let

\begin{equation*} \pi _x:\bigl (\tB _g(x,\rho ),\tilde x\bigr )\longrightarrow \bigl (B_g(x,\rho ),x\bigr ) \end{equation*}

be the Riemannian universal cover of the open ball. The point

x

is a

(\delta ,\rho )

-local rewinding Reifenberg point, abbreviated

(\delta ,\rho )

-LRRP, if

\begin{equation}\label {eq:lrrp} d_{GH}\bigl (B_{\pi _x^*g}(\tilde x,s),B_{\R ^n}(0,s)\bigr )<\delta s \qquad (0<s<\rho ),\tag{1.1} \end{equation}

for the pointed Gromov–Hausdorff distance, the centres of the balls being the base points. The manifold has uniform

(\delta ,\rho )

-LRRP if every point is a

(\delta ,\rho )

-LRRP. The point

x

is a

(\rho ,v)

-local rewinding volume point, abbreviated

(\rho ,v)

-LRVP, if

\begin{equation*} \vol _{\pi _x^*g}B_{\pi _x^*g}(\tilde x,\rho )\ge v. \end{equation*}

The terminology follows [HRW20; Ron26a; Ron26b]. LRRP controls the balls about $\tilde x$ in the local cover at every scale below $\rho$ , LRVP only the volume of the ball of radius $\rho$ . Under $\Ric _g\ge -(n-1)g$ , LRRP with $\delta \le \delta (n)$ implies LRVP with $v=v(n,\rho )>0$ by Colding’s volume convergence theorem [Col97]; the converse fails. Rong’s conjecture under the stronger condition reads as follows.

Conjecture ([Ron26b, Conjecture 0.6, case (0.6.1)]).Let $(M_i,g_i)$ be closed $n$ -manifolds. Assume that $\Ric _{g_i}\ge -(n-1)g_i$ and that all their points are $(\delta ,\rho )$ -LRRP points. Suppose that $(M_i,g_i)$ converges in the Gromov–Hausdorff topology to a compact metric space $X$ of positive diameter and that $\vol _{g_i}(M_i)\to 0$ . Then, for $i$ large, there is a singular nilpotent fibration $f_i:M_i\to X$ .

Rong uses the unpointed distance in (1.1), which changes $\delta$ by at most a factor of two (Section 2.1). Rong’s definition requires $\delta \le \delta (n)$ ; we prove the conjecture for $\delta \le \delta _1(n)$ and show that the conclusion fails in general for $\delta >11/12$ .

Under the weaker LRVP hypothesis the picture is different. The earlier proposals under a Ricci lower bound and uniform LRVP all expect, at a minimum, that the convergence is realized by continuous maps $f_i:M_i\to X$ whose approximation errors tend to zero. Case (0.6.2) of Conjecture 0.6 in [Ron26a] asks in addition for finite infranil-quotient fibres capturing the short loops. Open Problem 4.4 of [Ron26b] fixes $\eps ,\rho _0>0$ , writes $X_\eps$ for the $\eps$ -regular region and puts

\begin{equation*} X_0:=\{x\in X_\eps :d(x,X\setminus X_\eps )\ge \rho _0\}. \end{equation*}

A smooth infranil bundle is already available over $X_0$ , and the problem asks whether it is the restriction of such a map. Our examples show that under LRVP the continuous approximation itself need not exist, so both statements fail regardless of their requirements on the fibres or over $X_0$ .

1.1Main Results

Uniform LRRP supplies what ordinary short-time existence on a closed manifold lacks: control independent of the initial curvature.

Theorem 1.2 (Uniform Lifetime on Closed Manifolds). Fix $n\ge 2$ , $\rho >0$ and $\alpha \in (0,1)$ . There are $\delta _0=\delta _0(n,\alpha )>0$ and $T_0=T_0(n,\rho ,\alpha )>0$ with the following property. If $(M^n,g_0)$ is closed, $\Ric _{g_0}\ge -(n-1)g_0$ , and every point of $M$ is a $(\delta _0,\rho )$ -LRRP, then the maximal Ricci flow $g(t)$ with $g(0)=g_0$ satisfies

\begin{equation}\label {eq:main-bound} T_{\max }>T_0,\qquad \sup _M|\Rm _{g(t)}|\le \frac {\alpha }{t}\quad (0<t\le T_0).\tag{1.2} \end{equation}

The second result allows complete noncompact data with unbounded initial curvature.

Theorem 1.3 (Complete Manifolds). Fix $n\ge 2$ , $\rho >0$ and $\alpha \in (0,1)$ . There are constants $\delta _{\mathrm {nc}}\in (0,\delta _0]$ and $T_{\mathrm {nc}}\in (0,T_0]$ , depending only on $(n,\alpha )$ and $(n,\rho ,\alpha )$ , respectively, with the following property. If $(M^n,g_0)$ is complete, $\Ric _{g_0}\ge -(n-1)g_0$ , and every point of $M$ is a $(\delta _{\mathrm {nc}},\rho )$ -LRRP, then there is a complete Ricci flow $g(t)$ , smooth on $M\times [0,T_{\mathrm {nc}}]$ , with $g(0)=g_0$ and

\begin{equation*} \sup _M|\Rm _{g(t)}|\le \frac {\alpha }{t}\quad (0<t\le T_{\mathrm {nc}}). \end{equation*}

It is the only complete Ricci flow on

M\times [0,T_{\mathrm {nc}}]

with initial metric

g_0

whose curvature satisfies

\sup _M|\Rm _{g(t)}|\le A/t

for some constant

A

For noncompact $M$ the existence is new; the uniqueness recovers [Lee26, Theorem 1.1] by the same comparison argument. At positive time, Theorem 1.2 gives a bounded-curvature metric uniformly bi-Hölder to $g_0$ , so bounded-curvature collapsing theory applies. For a map $f:Z\to Z'$ between metric spaces put

\begin{equation*} \dis f:=\sup _{z_1,z_2\in Z}\bigl |d_{Z'}(f(z_1),f(z_2))-d_Z(z_1,z_2)\bigr |; \end{equation*}

$f$ is an $\eps$ -Gromov–Hausdorff approximation if $\dis f<\eps$ and $f(Z)$ is $\eps$ -dense in $Z'$ .

Theorem 1.4 (Singular Nilpotent Fibrations). For every $n\ge 2$ there is $\delta _1=\delta _1(n)>0$ with the following property. Let $\rho >0$ , and let $(M_i,g_i)$ be closed $n$ -manifolds with $\Ric _{g_i}\ge -(n-1)g_i$ and uniform $(\delta _1,\rho )$ -LRRP. Suppose that they converge in the Gromov–Hausdorff topology to a compact metric space $X$ of positive diameter, and that $\vol _{g_i}(M_i)\to 0$ . Then there are $\eps _i\to 0$ and, for $i$ large, continuous surjective maps $f_i:M_i\to X$ such that

(i): $f_i$ is an $\eps _i$ -Gromov–Hausdorff approximation from $(M_i,d_{g_i})$ to $X$ ;
(ii): every fibre of $f_i$ is an infranilmanifold of positive dimension;
(iii): $f_i$ is a singular nilpotent fibration: it is induced by an $O(n)$ -equivariant fibre bundle, with nilmanifold fibre and affine structure group, from the frame bundle of $M_i$ onto a closed manifold $Y_i$ with $Y_i/O(n)$ homeomorphic to $X$ .

This is case (0.6.1) of [Ron26b, Conjecture 0.6] for $\delta \le \delta _1(n)$ , in the precise form of Theorem 8.2 and with the qualification in Section 1.4. If the $M_i$ are moreover aspherical with fundamental groups of type $A_s$ , Corollary 8.4 shows that $X$ is homeomorphic to, and hence abstractly smoothable as, an aspherical manifold: the manifold and asphericity part of [Ron26b, Conjecture 0.7], without its quantitative Reifenberg conclusion for the original metric on $X$ .

Under the weaker condition, Nifa’s closed four-manifolds [Nif26] give limits onto which there are no continuous almost-isometries.

Theorem 1.5 (No Continuous Approximation). Let $(\Sigma _j,g_j)$ be the closed four-manifolds of [Nif26, Theorem 1.3], and let $\Sigma _\infty$ be their Gromov–Hausdorff limit. Let $\lambda _j\downarrow 0$ with $\lambda _j<2/3$ , and give $P_j:=\Sigma _j\times (\R /\lambda _j\Z )$ the product metric. Then the following hold.

(i): $\Ric _{P_j}\ge 0$ , and there is $v_*>0$ such that every point of every $P_j$ is a $(1,v_*)$ -LRVP.
(ii): $P_j\to \Sigma _\infty$ in the Gromov–Hausdorff topology, $\diam \Sigma _\infty >0$ , and $\vol (P_j)\to 0$ .
(iii): There are $\eps _0>0$ and $j_0$ such that $\dis f\ge \eps _0$ for every continuous map $f:P_j\to \Sigma _\infty$ with $j\ge j_0$ . In particular, for $\eps <\eps _0$ and $j\ge j_0$ there is no continuous $\eps$ -Gromov–Hausdorff approximation from $P_j$ to $\Sigma _\infty$ .
(iv): $\Sigma _\infty$ is not locally contractible, and it is not homeomorphic to the orbit space of a smooth action of a compact Lie group on a manifold.
(v): Every point of every $P_j$ is a $(\delta ,\rho )$ -LRRP for all $\delta >\frac {11}{12}$ and all $\rho >0$ , while for no $\delta \le 0.44$ and no $\rho >0$ do all the $P_j$ have uniform $(\delta ,\rho )$ -LRRP.

Part (iii) excludes every conjectured structure, each of which includes continuous Gromov–Hausdorff approximations with errors tending to zero.

Corollary 1.6 (The Statements under LRVP). The manifolds $P_j$ of Theorem 1.5 are closed five-manifolds with $\Ric \ge -4$ and uniform $(1,v_*)$ -LRVP that collapse to the compact space $\Sigma _\infty$ . For this sequence: (a) there is no singular nilpotent fibration $P_j\to \Sigma _\infty$ for $j$ large, so Theorem 1.4 fails with LRRP replaced by LRVP, or by $(\delta ,\rho )$ -LRRP with $\delta >\frac {11}{12}$ ; (b) case (0.6.2) of [Ron26a, Conjecture 0.6] fails; (c) [Ron26b, Open Problem 4.4] has a negative answer; and (d) [HRW20, Conjecture 3.3] fails whenever a singular fibration is, in particular, a continuous Gromov–Hausdorff approximation onto the limit with error tending to zero.

The Reifenberg constant of a sequence is the infimum of the $\delta$ for which some $\rho >0$ gives uniform $(\delta ,\rho )$ -LRRP. For $P_j$ it lies between $0.44$ and $\frac {11}{12}$ ; the upper bound holds for every product of a circle with a complete four-manifold of nonnegative Ricci curvature.

1.2Mechanism and Proof Strategy

Theorems 1.2 and 1.3 rest on a local statement for a Ricci flow that need not be complete, Proposition 3.1: an $A/t$ curvature bound on the initial ball $B_{g(0)}(p,4r)$ , with a Ricci lower bound and Euclidean closeness on $B_{g(0)}(p,2r)$ at time zero, improves at $p$ to $\alpha /t$ , $\alpha <A$ , for a time proportional to $r^2$ . Applied with $A=1$ on each local cover, a first-time argument gives the uniform lifetime (Section 3).

For the local statement, Reifenberg control and the initial Ricci bound give almost-Euclidean isoperimetry [Col97; CM20]; Hochard’s conformal completion [Hoc16; HW22] produces a complete metric that agrees with the original one on a smaller ball, and its Ricci flow has a $\beta /t$ bound with $\beta \ll \alpha$ by Wang’s pseudolocality theorem [Wan20] (Section 4). A localized version of Lee’s comparison through the Ricci–harmonic map heat flow [Lee26] transfers this bound to the incomplete lifted flow (Sections 5 and 6). The Ricci lower bound enters only at time zero, only the comparison flow is complete, and distinct local covers are never compared; the inequality between the conformal and original metrics replaces the one global distance-conversion step in Lee’s argument and avoids the patching issue noted in [Ron26b, Section 1.3].

For noncompact data, conformal metrics equal to $g_0$ on an exhaustion push the artificial boundaries to infinity; isoperimetry on the local covers survives at an adaptive scale near the collar, and local estimates through time zero [Che09; ST21] give a complete limit flow (Section 7). For Theorem 1.4, the distance distortion estimate of [HW22, Appendix A] identifies the smoothed and original limits bi-Hölderly, and the theory of Fukaya and Cheeger–Fukaya–Gromov supplies the fibrations (Section 8).

Theorem 1.5 (Section 9) is independent of the Ricci flow. The circle factor gives the rewinding-volume bound; a small-distortion map would have degree one over the regular part of $\Sigma _\infty$ and so wrap $\Sigma _j$ around infinitely many copies of $\CP ^2$ minus a ball, which would give infinitely many independent classes in the finite-dimensional space $H^2(\Sigma _j;\Z _2)$ ; the Reifenberg bounds follow from Bishop–Gromov and, at the accumulation point, from the conical tangent via the quantitative Borsuk–Ulam theorem [DS81; LMS23].

1.3Relation to Prior Work

The nilpotent structure under bounded sectional curvature is due to Fukaya [Fuk87; Fuk89] and Cheeger–Fukaya–Gromov [CFG92]; the programme under a Ricci lower bound is surveyed in [HRW20]. For manifold limits, Rong obtained affine nilpotent bundles without smoothing [Ron26a, Theorem B], [Ron26b, Theorem 0.12], and Huang and Wang [HW22] used uniform Ricci-flow smoothing under Euclidean or flat-orbifold models with a maximal-rank hypothesis. Theorem 1.2 imposes no condition on the collapsed limit.

Our analytic tools originate in Hochard’s conformal completion [Hoc16], the complete-flow limit of Simon–Topping [ST21], Lee’s comparison and uniqueness theory [Lee26], and Wang’s scale-invariant refinement [Wan20] of Perelman’s pseudolocality theorem [Per02, Theorem 10.1]; the entropy input follows from scalar-curvature and isoperimetric bounds [Wan18, Lemma 3.5].

Under LRVP, Rong’s six-dimensional example already excludes a singular nilpotent fibration but still admits continuous approximations [Ron26b, Example 4.2]; see Section 10.2. Our examples exclude even those.

1.4Scope and Organization

Theorems 1.2 to 1.4 rest on the Ricci–harmonic map method of [Lee26] and the pseudolocality theorem [Wan20, Theorem 1.2]. The local estimates we need are collected in Proposition 2.3; its part (iv) is proved in Section 2.4 from [Lee26, Lemma 2.5] and the local maximum principle of [LT22], and its other parts in Appendix A, using the short-time existence lemma of [BK22]. The localized comparison is proved in Section 5, and complete Type-I uniqueness is derived in Section 7. Corollary 8.4 also uses [Ron26b, Theorem A]. Theorem 1.5 and Corollary 1.6 are independent of the Ricci-flow results and conditional on the unrefereed construction [Nif26], whose required properties are listed in Section 9.1.

The constants $\delta _0$ , $\delta _{\mathrm {nc}}$ and $\delta _1$ are small and not computed. In Theorem 1.4 the frame-bundle diagram is formed with the metrics smoothed by Ricci flow, since a Ricci lower bound does not control the canonical frame-bundle metric; the identification $X\cong Y_i/O(n)$ is therefore bi-Hölder rather than isometric, although the $f_i$ are Gromov–Hausdorff approximations for the original metrics. We prove neither the quantitative Reifenberg regularity of the original limit metric in [Ron26b, Conjecture 0.7] nor case (0.6.2) of [Ron26b, Conjecture 0.6]; Section 10 discusses the remaining cases.

After the preliminaries, Sections 3 to 6 prove Theorem 1.2; the remaining principal results are proved in Sections 7 to 9.

2Preliminaries

We fix metric conventions, the local completeness available on local universal covers, and the analytic inputs of Wang and Lee.

2.1Conventions

Manifolds are smooth and connected, and “closed” means compact without boundary. Write $B_g(x,r)$ and $d_g$ for Riemannian balls and distance, and $B_Z(z,r)$ in a metric space. A regular domain is relatively compact, open, and smoothly bounded; $\omega _n$ is the volume of the Euclidean unit ball. Ricci flows satisfy $\partial _tg=-2\Ric _g$ . Constants display their dependencies when needed; unadorned $C,c$ depend only on the parameters of the current statement and may change by line.

A correspondence between metric spaces $(Z,d)$ and $(Z',d')$ is a subset $\mathcal R\subset Z\times Z'$ whose two projections are onto, and its distortion is

\begin{equation*} \dis \mathcal R:=\sup \bigl \{|d(z_1,z_2)-d'(z_1',z_2')|: (z_1,z_1'),(z_2,z_2')\in \mathcal R\bigr \}. \end{equation*}

The Gromov–Hausdorff distance is $d_{GH}(Z,Z')=\tfrac 12\inf _{\mathcal R}\dis \mathcal R$ [BBI01, Theorem 7.3.25]. For pointed spaces we use

\begin{equation*} d_{GH}\bigl ((Z,z),(Z',z')\bigr ):=\tfrac 12\inf \{\dis \mathcal R:(z,z')\in \mathcal R\}. \end{equation*}

This is at least the unpointed distance. For balls with displayed centres, including (1.1) and (3.3), $d_{GH}$ is pointed. If every point of $Z$ lies within $s$ of $z$ and a correspondence between $Z$ and $B_{\R ^n}(0,s)$ has distortion at most $\eps$ , every point related to $z$ has norm at most $\eps$ ; adding $(z,0)$ therefore at most doubles the distortion. Thus Rong’s unpointed form of (1.1) implies ours with $2\delta$ in place of $\delta$ .

2.2Local Universal Covers

The local covers are incomplete. For $\tilde z\in \tB _g(x,\rho )$ call $\rho _{\tilde z}:=\rho -d_{\pi _x^*g}(\tilde x,\tilde z)$ its depth. Positive depth supplies the following local substitute for completeness.

Lemma 2.1 (Local Completeness of the Cover). Let $(M^n,g)$ be a complete Riemannian manifold, $x\in M$ and $\rho >0$ , and let $\tilde z\in \tB _g(x,\rho )$ have depth $\rho _{\tilde z}>0$ .

(i): The exponential map of $\pi _x^*g$ at $\tilde z$ is defined on the open ball of radius $\rho _{\tilde z}$ in the tangent space.
(ii): For $0<r<\rho _{\tilde z}$ , the closed ball of radius $r$ about $\tilde z$ is compact, and each of its points is joined to $\tilde z$ by a minimizing geodesic.
(iii): If $\Ric _g\ge 0$ and $0<r_1<r_2<\rho _{\tilde z}$ , then $\vol B_{\pi _x^*g}(\tilde z,r_2)\le (r_2/r_1)^n\vol B_{\pi _x^*g}(\tilde z,r_1)$ .

Proof.For $z=\pi _x(\tilde z)$ , $d_g(x,z)\le d_{\pi _x^*g}(\tilde x,\tilde z)$ . Thus every $g$ -geodesic from $z$ of length below $\rho _{\tilde z}$ stays in $B_g(x,\rho )$ and lifts, proving (i). The local Hopf–Rinow proof [Car92, Chapter 7, Theorem 2.8] gives (ii), and the standard Jacobian comparison along these minimizing geodesics gives (iii).∎

2.3Local Entropy and Pseudolocality

For a metric $h$ on an $n$ -manifold, a relatively compact ball $B=B_h(x,r)$ and $s>0$ , the local entropy $\nu (B,h,s)$ of [Wan18] is the infimum of

\begin{equation*} -n-\frac n2\log (4\pi s') +\int _B\Bigl (s'\bigl (\Scal _h\psi ^2+4|\nabla \psi |^2\bigr )-2\psi ^2\log \psi \Bigr )\,d\vol _h \end{equation*}

over $s'\in (0,s]$ and $\psi \in W^{1,2}_0(B)$ with $\psi \ge 0$ and $\int _B\psi ^2\,d\vol _h=1$ . Wang’s pseudolocality theorem turns a lower bound for this entropy into a scale-invariant curvature bound; Section 4 uses it to give the comparison flow its Type-I bound.

Theorem 2.2 (Wang’s Pseudolocality Theorem [Wan20, Theorem 1.2]). For every positive $\beta$ smaller than $1/(100n)$ there is $\delta _W=\delta _W(n,\beta )>0$ with the following property. Let $h(t)$ , $0\le t\le T$ , be a complete Ricci flow with bounded curvature on an $n$ -manifold, and let $x$ be a point of the manifold. If

\begin{equation*} \nu \bigl (B_{h(0)}(x,\delta _W^{-1}\sqrt T),h(0),T\bigr )\ge -\delta _W^2, \end{equation*}

then

t\,|\Rm _{h(t)}|\le \beta

B_{h(t)}(x,\beta ^{-1}\sqrt t)

for every

t\in (0,T]

2.4Ricci–Harmonic Comparison Inputs

We use the Ricci–harmonic gauge method of [Lee26, Sections 2–3]. The following proposition states the local versions needed here; Appendix A supplies the existence and derivative proofs. Derivatives of a solution $\partial _tF=\Delta _{g(t),h(t)}F$ use $g(t)$ and $h(t)$ ; when $F_t$ is a diffeomorphism onto its image, $\widehat g=(F_t^{-1})^*g$ is Ricci–DeTurck relative to $h$ .

Proposition 2.3 (Local Comparison Estimates). All source and target manifolds have dimension $n$ . Let $\eps _{\mathrm {DT}}(n)\in (0,10^{-2}]$ be a threshold below which the discrepancy inequality (2.1) below holds, and set $\eps _{\mathrm L}(n):=\eps _{\mathrm {DT}}(n)/10$ .

(i): Local Existence. For every $\Lambda _0>0$ there is $T_{\mathrm {ex}}=T_{\mathrm {ex}}(n,\Lambda _0)>0$ with the following property. Let $U\Subset M$ be a smooth domain, let a possibly incomplete Ricci flow $g(t)$ be smooth on a neighbourhood of $\overline U\times [0,T]$ , let $N$ carry a complete Ricci flow $h(t)$ , and suppose, on $[0,T]$ , that for some $r>0$
$\begin{equation*} \sup _U|\Rm _g|+\sup _N|\Rm _h|\le r^{-2},\qquad \sup _U|\nabla \Rm _g|\le r^{-3}, \end{equation*}$
while all curvature derivatives of $h(t)$ are globally bounded on $[0,T]$ . If $f:U\to N$ is smooth on a neighbourhood of $\overline U$ and $|df|^2\le \Lambda _0$ , then the Ricci–harmonic map heat flow starting from $f$ exists, with $|dF|^2\le 10\Lambda _0$ , on
$\begin{equation*} U_r:=\{x\in U:B_{g(0)}(x,r)\Subset U\} \quad \hbox {for}\quad 0\le t<\min \{T,T_{\mathrm {ex}}r^2\}. \end{equation*}$
The proof is Lemma A.1.
(ii): Local Derivatives. Given $A,\Lambda _0>0$ and an integer $q\ge 0$ , there are $S_q=S_q(n,A,\Lambda _0,q)>0$ and $K_q=K_q(n,A,\Lambda _0,q)<\infty$ such that the following holds. If the Ricci flows $g(t),h(t)$ satisfy $B_{g(0)}(x,4r)\Subset M$ , the target flow is complete, and
$\begin{equation*} \sup _{B_{g(0)}(x,4r)}|\Rm _g|+\sup _N|\Rm _h|\le A/t, \qquad |dF|^2\le \Lambda _0, \end{equation*}$
where $F$ is a smooth Ricci–harmonic map heat flow on the radius- $4r$ source cylinder, then on $B_{g(0)}(x,r)$
$\begin{equation*} |\nabla ^j dF|^2\le K_qt^{-j}\qquad (0\le j\le q) \end{equation*}$
for $0<t\le \min \{T,S_qr^2\}$ ; see Lemma A.2.
(iii): Ricci–DeTurck Derivatives. Let $g$ , $h$ , $x$ , $r$ , $\Lambda _0$ and $F$ be as in (ii), and suppose in addition that every $F_t$ is a diffeomorphism onto its image with
$\begin{equation*} (1-\eps _{\mathrm L})g(t)\le F_t^*h(t)\le (1+\eps _{\mathrm L})g(t) \quad \hbox {on }B_{g(0)}(x,4r). \end{equation*}$
Then $\widehat g=(F_t^{-1})^*g$ satisfies, on $F_t(B_{g(0)}(x,r))$ and for $0<t\le \min \{T,S_qr^2\}$ ,
$\begin{equation*} |(\nabla ^h)^j\widehat g|_h\le K_q^{\mathrm {DT}}t^{-j/2}\qquad (1\le j\le q), \end{equation*}$
with $K_q^{\mathrm {DT}}=K_q^{\mathrm {DT}}(n,A,\Lambda _0,q)$ ; see Lemma A.3.
(iv): Ricci–DeTurck Decay. Given $A,K>0$ and an integer $L\ge 1$ there is a constant $T_L^{\mathrm {dec}}>0$ , depending only on $n$ , $A$ , $K$ and $L$ , with the following property. Let $h(t)$ be a smooth Ricci flow on $N$ with $B_{h(0)}(y,4r)\Subset N$ and $|\Rm _h|\le A/t$ there, and let $\widehat g$ be a smooth Ricci–DeTurck flow relative to $h$ on a neighbourhood of the closed reference cylinder $\overline {B_{h(0)}(y,4r)}\times [0,T]$ , with $\widehat g(0)=h(0)$ , satisfying there
$\begin{equation*} (1-\eps _{\mathrm {DT}})h\le \widehat g\le (1+\eps _{\mathrm {DT}})h, \qquad |(\nabla ^h)^2\widehat g|_h\le \frac {K}{t}. \end{equation*}$
Then, on $B_{h(0)}(y,r)$ and for $0<t\le \min \{T,T_L^{\mathrm {dec}}r^2\}$ ,
$\begin{equation*} |\widehat g-h|_h\le (t/r^2)^L, \qquad |\nabla ^h(\widehat g-h)|_h^2\le r^{-2}(t/r^2)^L. \end{equation*}$

Proof of part (iv). Rescale $r=1$ and write

\begin{equation*} H=\widehat g-h,\qquad v=|H|_h^2, \end{equation*}

so that

|(\nabla ^h)^2H|\le K/t

B_{h(0)}(y,4)

by hypothesis. The covariant discrepancy equation gives, for small

\eps _{\mathrm {DT}}

\begin{equation}\label {eq:discrepancy-decay-pde} (\partial _t-\Delta _h)v\le \mathcal L_0v,\qquad \mathcal L_0=C_n\bigl (|\Rm _h|+|(\nabla ^h)^2H|\bigr )\le b(n,A,K)/t;\tag{2.1} \end{equation}

see [Lee26, Lemma 2.5]; there the term

-|\nabla H|^2

absorbs the quadratic gradient terms, and replacing the operator of

\widehat g

\Delta _h

costs

C_n|H|^2|\nabla ^2H|

. Both flows are smooth on the closed cylinder, so

\mathcal L_0

is bounded there; only the bound

b/t

degenerates at

t=0

. After enlarging

b

, one also has

\Ric _h\le b/t

and

v\le b/t

for

t\le 1

. For each

z\in B_{h(0)}(y,3/2)

, the ball of radius

2

about

z

has closure in

B_{h(0)}(y,7/2)\Subset B_{h(0)}(y,4)

. Since

v(0)=0

, [LT22, Theorem 1.1], with an integer exponent

k>\max \{2J,b+1\}

, therefore gives uniformly

\begin{equation*} \sup _{B_{h(0)}(y,3/2)}|H|_h\le t^J \end{equation*}

for any prescribed integer

J

, on a time depending only on

n,A,K,J

For $x\in B_{h(0)}(y,1)$ , the shrinking-balls lemma [ST21, Lemma 2.2], applied to $B_{h(0)}(x,3/8)\Subset B_{h(0)}(y,3/2)$ , gives $B_{h(t)}(x,1/4)\Subset B_{h(0)}(y,3/2)$ for a common smaller time. Along a unit-speed $h(t)$ -geodesic of length $s=t^{(J+1)/2}\le 1/4$ , which exists because $B_{h(t)}(x,1/4)$ is relatively compact, express $H$ in a parallel frame and apply Taylor’s formula to each component. This gives

\begin{equation}\label {eq:gradient-decay-interpolation} |\nabla ^hH|(x,t)\le C_n\left (\frac {\sup |H|}{s}+s\sup |(\nabla ^h)^2H|\right ) \le C(n,A,K,J)t^{(J-1)/2}.\tag{2.2} \end{equation}

Choose

J=L+2

and decrease the time to absorb the spare factor of

t

in the square of (2.2). Restoring

r

gives the two bounds.∎

Section 5 applies these estimates on successive time slabs.

3The Local Improvement and the First-Time Argument

We state the local improvement for incomplete flows and derive Theorem 1.2; Sections 4 to 6 prove the local statement.

Proposition 3.1 (Local Type-I Improvement). Fix $n\ge 2$ , $A\ge 1$ and $0<\alpha <A$ . There are $\delta _*=\delta _*(n,A,\alpha )>0$ , $\kappa _*=\kappa _*(n,A,\alpha )>0$ , and $\Theta =\Theta (n,A,\alpha )\in (0,1)$ with the following property. Let $g(t)$ , $0\le t<T$ , be a smooth Ricci flow on an $n$ -manifold $U$ , let $p\in U$ and $r>0$ , and assume that

(a): the closed ball $\overline {B_{g(0)}(p,4r)}$ is compact;
(b): on $B_{g(0)}(p,4r)\times (0,T)$ ,
$\begin{equation}\label {eq:hyp-typeI} |\Rm _{g(t)}|\le \frac {A}{t};\tag{3.1} \end{equation}$
(c): on $B_{g(0)}(p,2r)$ ,
$\begin{equation}\label {eq:hyp-ricci} \Ric _{g(0)}\ge -(n-1)\kappa _*r^{-2}g(0);\tag{3.2} \end{equation}$
(d): $\begin{equation}\label {eq:hyp-gh} d_{GH}\bigl (B_{g(0)}(p,2r),B_{\R ^n}(0,2r)\bigr )\le \delta _*r.\tag{3.3} \end{equation}$

Then

\begin{equation}\label {eq:improved} |\Rm _{g(t)}|(p)\le \frac {\alpha }{t} \quad \bigl (0<t<\min \{T,\Theta r^2\}\bigr ),\tag{3.4} \end{equation}

and consequently

\begin{equation}\label {eq:improved-all} |\Rm _{g(t)}|(p)\le \frac {\alpha }{t}+\frac {A}{\Theta }\,r^{-2} \quad (0<t<T).\tag{3.5} \end{equation}

All hypotheses concern the initial ball, because no Ricci lower bound is assumed at positive time. Lifting the global flow to each local cover and applying the gain $\alpha <A$ while $|\Rm |\le 1/t$ gives the following first-time argument.

Proof of Theorem 1.2. Apply Proposition 3.1 with $A=1$ and the prescribed $\alpha$ . Let $\delta _*,\kappa _*,\Theta$ be its constants and put

\begin{equation*} r_0:=\min \left \{\frac {\rho }{16}, \frac {\sqrt {\kappa _*}}2\right \}, \qquad \delta _0:=\frac {\delta _*}{2}. \end{equation*}

Hamilton’s theorem [Ham82] starts the maximal flow for a positive, possibly nonuniform time.

Fix $x\in M$ and lift the flow to the fixed $g_0$ -cover $\tB _{g_0}(x,\rho )$ . Since $4r_0\le \rho /4$ , Lemma 2.1 makes the closed $4r_0$ -ball about $\tilde x$ compact, while (1.1) gives

\begin{equation*} d_{GH}\bigl (B_{\pi _x^*g_0}(\tilde x,2r_0), B_{\R ^n}(0,2r_0)\bigr ) <2\delta _0r_0\le \delta _*r_0. \end{equation*}

The choice

r_0^2\le \kappa _*/4

and the initial Ricci lower bound give

\begin{equation*} \Ric _{\pi _x^*g_0}\ge -(n-1)\kappa _*r_0^{-2}\pi _x^*g_0. \end{equation*}

Define

\begin{equation}\label {eq:TA} T_A:=\sup \left \{s<T_{\max }: t\sup _M|\Rm _{g(t)}|\le A \text { for every }0<t\le s\right \}.\tag{3.6} \end{equation}

Smoothness gives

T_A>0

. Before

T_A

, every lifted flow satisfies Proposition 3.1 with

p=\tilde x

and

r=r_0

, hence

\begin{equation}\label {eq:ft-improved} t\sup _M|\Rm _{g(t)}|\le \alpha <A \quad \bigl (0<t<\min \{T_A,\Theta r_0^2\}\bigr ).\tag{3.7} \end{equation}

T_A\le \Theta r_0^2

and

T_A<T_{\max }

, the strict margin and continuity extend (3.6); if

T_A=T_{\max }

, (3.7) bounds curvature on the final half-interval and continuation extends the flow. Thus

\begin{equation*} T_{\max }>\Theta r_0^2,\qquad T_A>\Theta r_0^2. \end{equation*}

One final application at every lifted point gives (1.2) with

T_0:=\frac 12\Theta r_0^2

.∎

4A Complete Comparison Flow

We construct the complete comparison flow using Hochard’s conformal completion [Hoc16; HW22] and Theorem 2.2. The rescaling-invariant lemmas are stated for reuse in Section 7.

4.1Constants

Set $\beta :=\min \{\alpha /32,1/(200n)\}$ . This is the Type-I constant of the comparison flow: it lies below Wang’s threshold $1/(100n)$ , and its gap to $\alpha$ absorbs the comparison error in (6.2). Let $\delta _W=\delta _W(n,\beta )$ be from Theorem 2.2, and put $\delta _P:=1-e^{-\delta _W^2/2}$ and $\eps _P:=\delta _W/2$ ; in Lemma 4.3 the isoperimetric deficit $\delta _P$ gives the entropy bound of Theorem 2.2, and $\eps _P$ fixes the lifetime of the comparison flow relative to the isoperimetric scale. Choose the parameter $\theta =\theta (n,\beta )>0$ of the conformal profile of Lemma 4.2 and then the isoperimetric deficit $\delta _I>0$ of the initial metric so that

\begin{equation}\label {eq:deltaI} \frac {1-\delta _I}{(1+2\theta )^{n(n-1)}}\ge 1-\frac 12\delta _P.\tag{4.1} \end{equation}

All these constants depend only on $n$ and $\alpha$ .

4.2Initial Isoperimetry

The first lemma converts the initial hypotheses into definite-scale isoperimetry.

Lemma 4.1 (Initial Isoperimetry). For every $\eta \in (0,1)$ there are constants $c(\eta )\in (0,1/10)$ , $\delta (\eta )>0$ and $\kappa (\eta )>0$ , depending only on $n$ and $\eta$ , with the following property. Let $g$ be a Riemannian metric on an $n$ -manifold, let $p$ be a point and $r>0$ , and suppose that the closed ball of radius $r/2$ about $p$ is compact and that

\begin{equation}\label {eq:iso-hyp} \Ric _g\ge -(n-1)\kappa (\eta )r^{-2}g\ \hbox { on }B_g(p,2r),\qquad d_{GH}\bigl (B_g(p,2r),B_{\R ^n}(0,2r)\bigr )\le \delta (\eta )r .\tag{4.2} \end{equation}

Then, with

R:=c(\eta )r

\begin{equation}\label {eq:scal-R} \Scal _g\ge -R^{-2}\quad \hbox {on }B_g(p,R),\tag{4.3} \end{equation}

and every regular domain

E\Subset B_g(p,R)

satisfies

\begin{equation}\label {eq:iso} \bigl (\area _g\partial E\bigr )^n \ge (1-\eta )n^n\omega _n \bigl (\vol _gE\bigr )^{n-1}.\tag{4.4} \end{equation}

Proof.Scale to $r=1$ . Let $\bar \varepsilon _{0,n},\bar \eta _{0,n},\bar \delta _{0,n},C_{0,n}$ be the constants of [CM20, Theorem 1.1] for lower Ricci bound $0$ and dimension $n$ . Choose $\eta _V\le \bar \eta _{0,n}$ , $r_V\le \min \{\bar \delta _{0,n},1/10\}$ , and $\varepsilon _V\le \bar \varepsilon _{0,n}$ so that

\begin{equation*} C_{0,n}(r_V+\eta _V+\varepsilon _V) \le 1-(1-\eta )^{1/n}. \end{equation*}

We use Colding’s local volume continuity [Col97, Main Lemma 2.1 and Theorem 0.1] in the following form. There are

\delta _C,\kappa _C\in (0,1)

and

r_C>1

such that

\Ric \ge -(n-1)\kappa _C

B(y,r_C)

and

d_{GH}(B(y,r_C),B_{\R ^n}(0,r_C))<\delta _C

imply

\vol B(y,1)\ge (1-\eta _V)\omega _n

; this is the form used in [HW22, proof of Lemma 2.4].

Set $c_C=(2r_C)^{-1}$ and $g_C=c_C^{-2}g$ . Restricting a correspondence in (4.2) to concentric balls and rescaling gives

\begin{equation*} d_{GH}\bigl (B_{g_C}(p,r_C),B_{\R ^n}(0,r_C)\bigr ) \le c_C^{-1}\Psi (\delta (\eta )\mid n), \qquad \Ric _{g_C}\ge -(n-1)\kappa (\eta )c_C^2g_C, \end{equation*}

where

\Psi (\delta \mid n)\to 0

with

\delta

. These balls correspond to relatively compact

g

-balls in

B_g(p,1/2)

. Choose

\delta (\eta ),\kappa (\eta )

so that this error is below

\delta _C

\kappa (\eta )c_C^2\le \kappa _C

, and

(n-1)\kappa (\eta )c_C^2\le \varepsilon _V

. Thus

\vol _{g_C}B_{g_C}(p,1)\ge (1-\eta _V)\omega _n

, and [CM20, Theorem 1.1] gives, for finite-perimeter

E\subset B_{g_C}(p,r_V)

\begin{equation*} \operatorname {Per}_{g_C}(E)\ge n\omega _n^{1/n} \bigl (1-C_{0,n}(r_V+\eta _V+(n-1)\kappa (\eta )c_C^2)\bigr ) \vol _{g_C}(E)^{(n-1)/n}. \end{equation*}

The parameter choice, the

n

th power, and rescaling prove (4.4) with

c(\eta )=c_Cr_V

. Finally (4.2) gives

\Scal _g\ge -n(n-1)\kappa (\eta )

; decreasing

\kappa (\eta )

until

n(n-1)\kappa (\eta )\le c(\eta )^{-2}

proves (4.3).∎

For $\eta =\delta _I$ , fix

\begin{equation}\label {eq:cI} c_I:=c(\delta _I),\qquad \delta _*:=\delta (\delta _I),\qquad \kappa _*:=\kappa (\delta _I),\tag{4.5} \end{equation}

the constants of Proposition 3.1, all depending only on $n,\alpha$ .

4.3Conformal Completion

The next conformal change is supported outside a smaller ball and nearly preserves isoperimetry; its domain is a superlevel-set component of the cutoff.

Lemma 4.2 (Conformal Completion). For every $C_\chi \ge 1$ there are constants $c_H\in (0,1/8]$ and $C_H>0$ , depending only on $n$ , $\beta$ and $C_\chi$ , with the following property. Let $g$ be a Riemannian metric on an $n$ -manifold, let $p$ be a point and $R>0$ , and suppose that the closed ball of radius $3R/4$ about $p$ is compact, that (4.3) holds, that (4.4) holds with $\eta =\delta _I$ , and that there is a smooth function $\chi$ with values in $[0,1]$ , supported in $B_g(p,3R/4)$ and equal to one on $B_g(p,R/3)$ , such that

\begin{equation}\label {eq:cutoff} R|\nabla \chi |+R^2|\Delta \chi |\le C_\chi .\tag{4.6} \end{equation}

Then there are an open set

\Omega

with

\begin{equation*} B_g(p,R/4)\Subset \Omega \Subset B_g(p,3R/4) \end{equation*}

and a smooth function

u\ge 0

\Omega

such that

h_0:=e^{2u}g

has the following properties, where

r_H:=c_HR

(i): $(\Omega ,h_0)$ is complete, $h_0\ge g$ , and $h_0=g$ on $B_g(p,R/4)$ ;
(ii): $h_0$ has bounded curvature;
(iii): $\Scal _{h_0}\ge -r_H^{-2}$ ;
(iv): every regular domain $E$ compactly contained in an $h_0$ -ball of radius $r_H$ satisfies
$\begin{equation}\label {eq:iso-P} \bigl (\area _{h_0}\partial E\bigr )^n \ge (1-\delta _P)n^n\omega _n\bigl (\vol _{h_0}E\bigr )^{n-1}.\tag{4.7} \end{equation}$

Proof.Choose a regular value $a\in (1/4,1/2)$ of $\chi$ and let $\Omega$ be the component of $\{\chi >a\}$ containing $B_g(p,R/3)$ ; it has smooth compact closure in $B_g(p,3R/4)$ . Fix a smooth decreasing $\upsilon _0:[0,1]\to [0,1]$ with $\upsilon _0=0$ near $1$ , $\upsilon _0(0)=1$ , $\upsilon _0'(0)<0$ , and $\upsilon _0(s)<1$ for $s>0$ , and set

\begin{equation*} \varphi :=\upsilon _0\circ \frac {\chi -a}{1-a}. \end{equation*}

Then

\varphi =0

B_g(p,R/3)

0\le \varphi <1

\varphi \to 1

\partial \Omega

, and (4.6) gives

\begin{equation}\label {eq:phi-bounds} R|\nabla \varphi |+R^2|\Delta \varphi |\le C_\varphi ,\tag{4.8} \end{equation}

where

C_\varphi =C_\varphi (C_\chi )

. The smooth extensions of

g,\varphi

past

\overline \Omega

have finite curvature and Hessian norms there.

Let $w_\theta :[0,1)\to [0,\infty )$ be Hochard’s conformal profile [Hoc16, proof of Lemma 6.2], smoothed as in [HW22, equations (2.3)–(2.5)]: it is nondecreasing and convex, vanishes on $[0,1-3\theta /2]$ , tends to infinity at $1$ , and satisfies

\begin{equation}\label {eq:w-deriv} e^{-w_\theta }w_\theta '\le C\theta ^{-1},\qquad e^{-2w_\theta }\bigl (w_\theta ''+(w_\theta ')^2\bigr ) \le C\theta ^{-2}.\tag{4.9} \end{equation}

For

1-3\theta /2<s<1

and

b=\theta (1-s)/4

, it also satisfies

\begin{equation}\label {eq:w-collar} e^{w_\theta (s+b)-w_\theta (s-b)} \le 1+2\theta ,\qquad b e^{w_\theta (s-b)}\ge c\theta ^2.\tag{4.10} \end{equation}

Set

u=w_\theta \circ \varphi

and

h_0=e^{2u}g

. The second estimate in (4.10) and (4.8) make every escaping curve have infinite

h_0

-length. Thus

h_0

is complete, dominates

g

, and equals

g

B_g(p,R/4)

. The conformal scalar-curvature formula, (4.3), (4.8), and (4.9) give

\begin{equation*} \Scal _{h_0}\ge -C_HR^{-2}. \end{equation*}

The full-curvature formula and the finite bounds on

\overline \Omega

give bounded sectional curvature.

Call the set $\{1-3\theta /2<\varphi <1\}$ , where $u>0$ , the collar. For a point $y$ of the collar put $s=\varphi (y)$ and $b=\theta (1-s)/4$ . The calculation of [HW22, equations (2.23)–(2.33)], or the same first-exit argument using (4.8)–(4.10), gives, for $c_{\mathrm {col}}\le c\theta ^2/C_\varphi$ ,

\begin{equation*} B_{h_0}(y,c_{\mathrm {col}}R)\subset \varphi ^{-1}(s-b,s+b),\qquad e^{2w_\theta (s-b)}g\le h_0 \le e^{2w_\theta (s+b)}g, \end{equation*}

where the metric inequality holds on the displayed ball, which lies in

B_g(p,R)

in the original metric. Comparing boundary area and volume with the two conformal factors, (4.4) and (4.10) give, for every regular domain

E

compactly contained in that ball,

\begin{equation}\label {eq:iso-h0} \frac {\bigl (\area _{h_0}\partial E\bigr )^n}{\bigl (\vol _{h_0}E\bigr )^{n-1}}\ge (1-\delta _I)(1+2\theta )^{-n(n-1)}n^n\omega _n \ge (1-\tfrac 12\delta _P)n^n\omega _n >(1-\delta _P)n^n\omega _n.\tag{4.11} \end{equation}

The middle inequality is (4.1). Set

c_H:=\min \{c_{\mathrm {col}}/2,1/8,C_H^{-1/2}\}

. Then (iii) holds. An

r_H

-ball contained in

\{u=0\}

satisfies (4.4); otherwise it meets the collar and lies in a

2r_H

-ball to which (4.11) applies. Domain monotonicity proves (iv).∎

The curvature bound in (ii) is qualitative; the next lemma uses it only to start the Ricci flow of $h_0$ .

4.4The Comparison Flow

We now apply Theorem 2.2 at every point.

Lemma 4.3 (Complete Comparison Flow). Let $h_0$ be a complete Riemannian metric with bounded curvature on an $n$ -manifold $\Omega$ , and let $r_H>0$ . Suppose that $\Scal _{h_0}\ge -r_H^{-2}$ and that every regular domain compactly contained in an $h_0$ -ball of radius $r_H$ satisfies (4.7). Then $h_0$ has a complete bounded-curvature Ricci flow $h(t)$ on $0\le t\le T_H:=(\eps _Pr_H)^2$ , and

\begin{equation}\label {eq:selected-bound} |\Rm _{h(t)}|\le \frac {\beta }{t} \quad (0<t\le T_H).\tag{4.12} \end{equation}

Proof.Shi [Shi89] starts a complete bounded-curvature flow. Let $0<T\le T_H$ precede its maximal time. At each $x$ , the scalar and isoperimetric hypotheses and [Wan18, Lemma 3.5] give

\begin{equation*} \nu \bigl (B_{h_0}(x,r_H),h_0,s\bigr ) \ge \log (1-\delta _P)-sr_H^{-2}\qquad (s>0). \end{equation*}

Since

\delta _W^{-1}\sqrt T\le r_H/2

and

\nu

does not decrease when the ball shrinks [Wan18, Proposition 2.1], the definitions of

\delta _P,\eps _P

give

\begin{equation*} \nu \bigl (B_{h_0}(x,\delta _W^{-1}\sqrt T),h_0,T\bigr ) \ge -\tfrac 12\delta _W^2-\eps _P^2=-\tfrac 34\delta _W^2>-\delta _W^2 . \end{equation*}

Thus Theorem 2.2 applies at every point and gives

\begin{equation}\label {eq:selected-T} |\Rm _{h(t)}|\le \frac {\beta }{t} \quad (0<t\le T).\tag{4.13} \end{equation}

If the maximal time were at most

T_H

, letting

T

approach it would bound curvature on its final half and Shi continuation would extend the flow. Hence the flow reaches

T_H

, and (4.13) becomes (4.12).∎

5Localized Comparison

The comparison flow of Section 4 must be compared with the given flow of Proposition 3.1. The two agree initially only on a ball, and only the comparison flow is complete. The following lemma is the local form of Lee’s Ricci–harmonic map comparison argument [Lee26] that this requires. It is the main analytic step of the paper: on a time interval proportional to $\ell ^2$ , where $\ell$ is the radius scale of the lemma, the given flow is, after a change of gauge, a perturbation of the comparison flow that vanishes to any prescribed order at the initial time.

Lemma 5.1 (Localized Type-I Comparison). Let $N\subset U$ be open subsets of one smooth manifold, with equality allowed, let $g(t)$ be a Ricci flow on $U\times [0,T]$ , let $p\in U$ and $\ell >0$ , and put $D_j:=B_{g(0)}(p,j\ell )$ . Suppose $D_{64}\Subset N$ , and let $h(t)$ , $0\le t\le T$ , be a complete smooth Ricci flow on $N$ . Assume

\begin{equation}\label {eq:initial-order} h(0)=g(0)\ \hbox {on }D_{64},\qquad h(0)\ge g(0) \ \hbox {on }N,\tag{5.1} \end{equation}

where

h(0)\ge g(0)

is an inequality of quadratic forms, that is,

h(0)(\xi ,\xi )\ge g(0)(\xi ,\xi )

for every tangent vector

\xi

at a point of

N

, and

\begin{equation}\label {eq:comp-curv} \sup _{D_{64}}|\Rm _{g(t)}|+\sup _N|\Rm _{h(t)}| \le \frac {A_*}{t}.\tag{5.2} \end{equation}

For every integer

L\ge 1

there are constants

\tau _L>0

C_{\mathrm {disp}}

and

C_L<\infty

, depending only on

n

A_*

and

L

, with the following property. There is a Ricci–harmonic map heat flow

F

from

D_8\times [0,\min \{T,\tau _L\ell ^2\})

N

with

F_0=\mathrm {id}

such that, for

0<t<\min \{T,\tau _L\ell ^2\}

, the map

F_t

is a diffeomorphism of

D_8

onto its image and

\begin{equation}\label {eq:comp-disp} d_{g(0)}(F_t(x),x)\le C_{\mathrm {disp}}\sqrt t,\qquad D_4\subset F_t(D_8)\subset D_{16}.\tag{5.3} \end{equation}

Moreover, for the dimensional constant

\eps _*:=\eps _{\mathrm L}(n)

fixed in Proposition 2.3,

\begin{equation}\label {eq:comp-equiv} (1-\eps _*)g(t)\le F_t^*h(t)\le (1+\eps _*)g(t) \quad \hbox {on }D_8.\tag{5.4} \end{equation}

\widehat g=(F_t^{-1})^*g

and

H=\widehat g-h

, then on

D_2

\begin{equation}\label {eq:comp-H} |H|_h\le C_L(t/\ell ^2)^L,\qquad |\nabla ^hH|_h^2\le C_L\ell ^{-2}(t/\ell ^2)^L,\tag{5.5} \end{equation}

and on

D_1

\begin{equation}\label {eq:comp-Rm} t|\Rm _{\widehat g}-\Rm _h|_h \le C_L(t/\ell ^2)^{L/2}.\tag{5.6} \end{equation}

The proof occupies the rest of the section. We rescale so that $\ell =1$ and keep the notation of the lemma. After fixing the constants and the induction hypothesis (Section 5.1), we prove the step from one time slab to the next as a separate statement (Lemma 5.2), and then run the induction and pass from the gauge estimates to the curvature comparison (Section 5.3). The construction follows [Lee26, Sections 2–3], with the local estimates of Proposition 2.3 in place of global ones. No global bound on the initial curvature of $h$ is needed: the initial Dirichlet solution uses only a compact source and target neighbourhood, and every later extension starts at a positive time, when completeness and the Type-I bound give global Shi estimates for $h$ .

5.1Constants and the Induction Hypothesis

The constants are fixed in the following order; each depends only on $n$ , $A_*$ and $L$ , and on those chosen before it. With $\eps _*=\eps _{\mathrm L}(n)$ as in Lemma 5.1, set $\Lambda _0:=2n$ and $\Lambda _1:=10\Lambda _0$ . Put $a_*:=c(n)A_*$ , so that integration of $|\partial _th|_h\le c(n)A_*/t$ gives

\begin{equation}\label {eq:growth} |\xi |_{h(t)}\le (t/s)^{a_*}|\xi |_{h(s)},\qquad 0<s\le t,\tag{5.7} \end{equation}

for tangent vectors $\xi$ , and choose an integer $m>2a_*+2$ ; it is the decay exponent in (5.20) below, and the bound on $m$ makes the first integral in (5.21) converge.

A time slab beginning at a time $t_k>0$ is handled with the two scales

\begin{equation*} \sigma _k:=c_*\sqrt {t_k},\qquad \sigma _k':=\frac {\Lambda }{8}\sqrt {t_k}, \end{equation*}

whose coefficients are fixed as follows. Choose $c_*=c_*(n,A_*)\in (0,1]$ so that the curvature and first-curvature-derivative bounds (5.13) below are at most $\sigma _k^{-2}$ and $\sigma _k^{-3}$ on the slab. The metric equivalence (5.10) imposed in the induction gives $|dF|^2\le n(1+\eps _*)<\Lambda _0$ . Choose $\mu =\mu (n,A_*)\in (0,1]$ with

\begin{equation}\label {eq:mu-choice} \mu <T_{\mathrm {ex}}(n,\Lambda _0)c_*^2,\qquad C_{\mathrm {bar}}\mu \le \tfrac 23\eps _*,\tag{5.8} \end{equation}

where $C_{\mathrm {bar}}=C_{\mathrm {bar}}(n,A_*)$ is the constant of the tensor barrier (5.16) below. The first inequality is the time restriction of Proposition 2.3(i) at the scale $\sigma _k$ , which also supplies the bound $|dF|^2\le \Lambda _1$ on the new slab. Then choose $\Lambda =\Lambda (n,A_*,L)\ge 1$ so large that

\begin{equation}\label {eq:Lambda-choice} \left (\frac {\Lambda }{8}\right )^2 \min \bigl \{S_2(n,A_*,\Lambda _1,2),\, T_m^{\mathrm {dec}}(n,A_*,K_2^{\mathrm {DT}},m)\bigr \}\ge 2, \qquad K_2^{\mathrm {DT}}:=K_2^{\mathrm {DT}}(n,A_*,\Lambda _1,2),\tag{5.9} \end{equation}

so that part (ii) of Proposition 2.3 at the scale $\sigma _k'$ covers the full interval $[0,t_{k+1}]$ whenever $t_{k+1}\le 2t_k$ , and parts (iii) and (iv), with exponent $m$ and scale $\sigma _k'$ , cover $[0,t_k]$ . The constant $\Lambda$ is increased at four further places in the proof of Lemma 5.2, each time by a quantity depending only on $n$ , $A_*$ and $L$ . Finally, the times $\tau '$ and $\tau _L$ are fixed in Lemma 5.2 and in its application.

For $t_k>0$ and $\mathfrak r_k>0$ put

\begin{equation*} B_{k,j}:=B_{g(0)}\bigl (p,\mathfrak r_k-j\Lambda \sqrt {t_k}\bigr ) \quad (j\ge 0). \end{equation*}

We say that a Ricci–harmonic map heat flow $F$ satisfies $(\mathrm {Ind}_k)$ for the data $(\mathfrak r_k,t_k)$ if $F$ is defined on $B_{k,0}\times [0,t_k]$ with $F_0=\mathrm {id}$ and has the following three properties there.

(I): $F_t$ is a diffeomorphism onto its image;
(II): $\begin{equation}\label {eq:ind-equiv} (1-\eps _*)g(t)\le F_t^*h(t)\le (1+\eps _*)g(t);\tag{5.10} \end{equation}$
(III): for $x\in B_{k,5}$ and $0\le t\le t_k$ ,
$\begin{equation}\label {eq:ind-disp} \begin {split} d_{g(0)}(x,F_t(x))&\le \frac 14\Lambda \sqrt {t_k},\\ B_{g(0)}(x,\Lambda \sqrt {t_k}) &\subset F_t\bigl (B_{g(0)}(x,2\Lambda \sqrt {t_k})\bigr ). \end {split}\tag{5.11} \end{equation}$

Figure 1 shows the regions that enter: panel (a) the domains $B_{k,0}\times [0,t_k]$ of the map flow along the induction, and panel (b) the step from $t_k$ to $t_{k+1}$ .

Figure 1. The regions in the proof of Lemma 5.1 , after the rescaling $\ell =1$ (schematic, not to scale).
(a) Space-time. The map flow $F_t$ is constructed on the shaded staircase: the initial Dirichlet flow lives on $D'$ up to the time $t_0$ , and at each time $t_k=(1+\mu )^kt_0$ the radius of the domain drops from $\mathfrak r_k$ to $\mathfrak r_{k+1}=\mathfrak r_k-6\Lambda \sqrt {t_k}$ . The total decrease of the
radius up to the time $\tau _L$ is less than $7$ , so the domain always contains the ball $D_8$ of the statement (dark).
The equality $h(0)=g(0)$ is used only inside $D_{32}$ , and the $33$ units between the staircase and $\partial D_{64}$ contain the parabolic balls
of the local Shi estimates. (b) The step from $(\mathrm {Ind}_k)$ to $(\mathrm {Ind}_{k+1})$ on the slab $[t_k,t_{k+1}]$ . The radius $6\Lambda \sqrt {t_k}$ lost in the step is spent in
margins of width $\Lambda \sqrt {t_k}$ , and each property holds on the ball below the corresponding line. — Figure 1. The regions in the proof of Lemma 5.1, after the rescaling $\ell =1$ (schematic, not to scale). (a) Space-time. The map flow $F_t$ is constructed on the shaded staircase: the initial Dirichlet flow lives on $D'$ up to the time $t_0$ , and at each time $t_k=(1+\mu )^kt_0$ the radius of the domain drops from $\mathfrak r_k$ to $\mathfrak r_{k+1}=\mathfrak r_k-6\Lambda \sqrt {t_k}$ . The total decrease of the radius up to the time $\tau _L$ is less than $7$ , so the domain always contains the ball $D_8$ of the statement (dark). The equality $h(0)=g(0)$ is used only inside $D_{32}$ , and the $33$ units between the staircase and $\partial D_{64}$ contain the parabolic balls of the local Shi estimates. (b) The step from $(\mathrm {Ind}_k)$ to $(\mathrm {Ind}_{k+1})$ on the slab $[t_k,t_{k+1}]$ . The radius $6\Lambda \sqrt {t_k}$ lost in the step is spent in margins of width $\Lambda \sqrt {t_k}$ , and each property holds on the ball below the corresponding line.

5.2The Step from One Slab to the Next

The next lemma extends the map flow across one time slab and recovers the induction hypothesis on a domain shrunk by six margins. Its proof is the core of the argument: the extension uses the local existence statement, a tensor barrier gives a rough metric equivalence and injectivity on the new slab, and the decay of the Ricci–DeTurck discrepancy on the earlier interval sharpens the equivalence back to (5.10).

Lemma 5.2 (One-Slab Extension). Let $g$ , $h$ , $N$ and $p$ satisfy the hypotheses of Lemma 5.1 with $\ell =1$ , and let the constants be as in Section 5.1. There is $\tau '=\tau '(n,A_*,L)\in (0,1]$ with the following property. Let $0<t_k\le \tau '$ and $24\le \mathfrak r_k\le 31$ , let $t_k<t_{k+1}\le (1+\mu )t_k$ with $t_{k+1}<T$ , and put $\mathfrak r_{k+1}:=\mathfrak r_k-6\Lambda \sqrt {t_k}$ . If $F$ satisfies $(\mathrm {Ind}_k)$ for the data $(\mathfrak r_k,t_k)$ , then $F$ extends to a Ricci–harmonic map heat flow on $B_{k,1}\times [0,t_{k+1}]$ that satisfies $(\mathrm {Ind}_{k+1})$ for the data $(\mathfrak r_{k+1},t_{k+1})$ .

Proof.Extension to the Next Slab. Since $\mathfrak r_k\le 31$ , one has

\begin{equation*} \operatorname {dist}_{g(0)}(B_{k,0},U\setminus D_{64})\ge 33. \end{equation*}

The shrinking-balls lemma [ST21, Lemma 2.2], under the upper Ricci bound implied by (5.2), gives a dimensional constant

C_n

such that, for

c>0

and

0\le t\le t_{k+1}

\begin{equation}\label {eq:shrink} B_{g(t)}(x,c\sqrt {t_k}) \subset B_{g(0)} \!\left (x,(c+C_n\sqrt {A_*})\sqrt {t_k}\right )\tag{5.12} \end{equation}

whenever the ball on the right lies in

D_{64}

. Put

\Lambda ':=1+C_n\sqrt {A_*}

and require

\tau '\le (33/\Lambda ')^2

. Then, for

x\in B_{k,0}

and

t_k/2\le t\le t_{k+1}

, the ball

B_{g(t)}(x,\sqrt {t_k})

lies in

D_{64}

, where the Type-I bound holds. Local Shi estimates for

g

on these parabolic neighbourhoods, together with complete Shi estimates for

h

, give for

t_k\le t\le t_{k+1}

, on

B_{k,0}

for the source and globally for the target,

\begin{equation}\label {eq:shi} |\nabla ^j\Rm _g|+|\nabla ^j\Rm _h| \le C_jt_k^{-1-j/2}.\tag{5.13} \end{equation}

Choose a smooth domain

B_k'

with

\begin{equation*} B_{g(0)}\bigl (p,\mathfrak r_k-\tfrac 12\Lambda \sqrt {t_k}\bigr ) \Subset B_k'\Subset B_{k,0}. \end{equation*}

Apply Proposition 2.3(i) to the time-shifted flows

g(t_k+\cdot )

and

h(t_k+\cdot )

, at the scale

\sigma _k

, with initial map

F_{t_k}

and source domain

B_k'

. Its curvature and first-curvature-derivative hypotheses hold on

B_k'

by (5.13), while (5.10) bounds

|dF_{t_k}|

; the source metric and

F_{t_k}

are smooth on a neighbourhood of

\overline {B_k'}

, since

\overline {B_k'}\subset B_{k,0}

; and (5.13) gives the global bounds for the curvature derivatives of

h

on the new slab. Increase

\Lambda

, if necessary, so that

\Lambda \ge 2(c_*+C_n\sqrt {A_*})

. Then (5.12) gives

\begin{equation*} B_{g(t_k)}(x,\sigma _k)\Subset B_k' \quad \hbox {for }x\in B_{k,1}. \end{equation*}

Hence the corresponding interior set contains

B_{k,1}

. Since

t_{k+1}-t_k\le \mu t_k<T_{\mathrm {ex}}\sigma _k^2

by (5.8) and

t_{k+1}<T

, the proposition extends

F

B_{k,1}\times [0,t_{k+1}]

, with

|dF|^2\le \Lambda _1

on the new slab; the extended map is smooth, because the two pieces agree at

t_k

and the equation determines all their time derivatives there.

Lower Bound and Injectivity. It remains to bound $dF$ from below and to prove injectivity on a smaller domain. Write $\check h(s):=F_s^*h(s)$ . For $x\in B_{k,2}$ one has $\overline {B_{g(0)}(x,4\sigma _k')}\Subset B_{k,1}$ . Hence Proposition 2.3(ii), applied on that ball, and (5.9) give on $B_{k,2}\times [t_k,t_{k+1}]$

\begin{equation}\label {eq:dF} |dF|\le C,\qquad |\nabla dF|\le Ct_k^{-1/2},\qquad |\nabla ^2dF|\le Ct_k^{-1}.\tag{5.14} \end{equation}

In particular, since

\partial _sF=\operatorname {tr}_{g(s)}\nabla dF

\begin{equation}\label {eq:dtF} |\partial _sF|_{h(s)}\le Ct_k^{-1/2},\qquad |\nabla (\partial _sF)|_{h(s)}\le Ct_k^{-1}.\tag{5.15} \end{equation}

These estimates supply a pointwise tensor barrier. There is $C_{\mathrm {bar}}=C_{\mathrm {bar}}(n,A_*)$ such that, if $0\le \eps <1$ and on $B_{k,2}$ at time $t_k$

\begin{equation*} (1-\eps )g(t_k)\le \check h(t_k)\le (1+\eps )g(t_k), \end{equation*}

then, for

t_k\le s\le t_{k+1}

satisfying

C_{\mathrm {bar}}(s-t_k)/t_k<1-\eps

\begin{equation}\label {eq:barrier} \left (1-\eps -C_{\mathrm {bar}}\frac {s-t_k}{t_k}\right )g(s) \le \check h(s)\le \left (1+\eps +C_{\mathrm {bar}}\frac {s-t_k}{t_k}\right )g(s)\tag{5.16} \end{equation}

B_{k,2}

. The time restriction keeps the lower scalar coefficient

b_-(s):=1-\eps -C_{\mathrm {bar}}(s-t_k)/t_k

positive; the upper coefficient is automatically positive. To prove the lower bound, fix

x\in B_{k,2}

. At a first null vector

\xi \in T_xU

, normalized by

g(s)(\xi ,\xi )=1

, of

\check h(s)-b_-(s)g(s)

, differentiating the tensor at fixed

x

gives

\begin{equation}\label {eq:contact} \begin {split} \partial _s\bigl [\check h(s)-b_-(s)g(s)\bigr ](\xi ,\xi ) &=2h\bigl (dF(\xi ),D_\xi \partial _sF\bigr ) -2\Ric _h\bigl (dF(\xi ),dF(\xi )\bigr )\\ &\quad -b_-'(s)+2b_-(s)\Ric _g(\xi ,\xi ). \end {split}\tag{5.17} \end{equation}

At contact

|dF(\xi )|_h^2=b_-(s)

; (5.2), (5.14)–(5.15), and

s\asymp t_k

bound every term except

-b_-'(s)

in absolute value by

C(n,A_*)t_k^{-1}

. Choosing

C_{\mathrm {bar}}>C(n,A_*)

makes (5.17) positive, contrary to first contact. Applying the same calculation to

b_+(s)g(s)-\check h(s)

, where

b_+(s)=1+\eps +C_{\mathrm {bar}}(s-t_k)/t_k

, proves the upper bound. Replacing

\eps

temporarily by

\eps +\eps '

and letting

\eps '\downarrow 0

handles non-strict initial inequalities.

Apply (5.16) with $\eps =\eps _*$ . Since $C_{\mathrm {bar}}(s-t_k)/t_k\le C_{\mathrm {bar}}\mu \le \frac 23\eps _*$ by (5.8), this gives

\begin{equation}\label {eq:rough} (1-3\eps _*)g(s)\le \check h(s)\le (1+3\eps _*)g(s)\tag{5.18} \end{equation}

B_{k,2}\times [t_k,t_{k+1}]

, so

F_s

is a local diffeomorphism there. It is injective on the smaller set

B_{k,3}

. Indeed, suppose

F_{s_0}(x_1)=F_{s_0}(x_2)=y_0

for

x_1,x_2\in B_{k,3}

and

s_0\in (t_k,t_{k+1}]

. Local invertibility defines backward trajectories by

\begin{equation}\label {eq:traj} \dot x_i(s)=-(dF_s|_{x_i(s)})^{-1} (\partial _sF_s)(x_i(s)),\qquad x_i(s_0)=x_i.\tag{5.19} \end{equation}

Here

dF_s

is the spatial differential, and

\partial _sF_s

is the time derivative at a fixed source point. The chain rule gives

\frac {d}{ds}[F_s(x_i(s))]=0

, so

F_s(x_i(s))=F_{s_0}(x_i)=y_0

along each trajectory. As long as a trajectory remains in

B_{k,2}

, (5.15) and (5.18) give

\begin{equation*} |\dot x_i(s)|_{g(s)}\le Ct_k^{-1/2}. \end{equation*}

The Type-I bound compares

g(s)

and

g(t_k)

uniformly on this slab. After integration, the local distance-distortion estimate between

g(t_k)

and

g(0)

gives

\begin{equation*} d_{g(0)}(x_i(s),x_i(s_0))\le C_1(n,A_*,\mu )\sqrt {t_k}. \end{equation*}

Choose

\Lambda >C_1

. The margin of width

\Lambda \sqrt {t_k}

between

B_{k,3}

and the boundary of

B_{k,2}

then prevents either trajectory from exiting before

t_k

. At

t_k

the induction hypothesis gives

x_1(t_k)=x_2(t_k)

, and uniqueness for the ODE (5.19) gives

x_1=x_2

. Thus

F_s|_{B_{k,3}}

is a diffeomorphism onto its image.

Sharpening and Displacement. It remains to sharpen the rough estimate (5.18) and recover (5.10)–(5.11). Fix $x\in B_{k,5}$ . By the induction hypothesis (5.11), for every $s\le t_k$ ,

\begin{equation*} B_{g(0)}(x,\Lambda \sqrt {t_k}) \subset F_s\bigl (B_{g(0)}(x,2\Lambda \sqrt {t_k})\bigr ). \end{equation*}

Hence

\widehat g_s=(F_s^{-1})^*g(s)

is defined on this fixed target ball. It lies in

D_{32}\Subset D_{64}

, where

\widehat g_0=h(0)

. For a fixed

s\le t_k

put

y=F_s(x)

. The displacement estimate of

(\mathrm {Ind}_k)

gives

d_{g(0)}(x,y)\le \Lambda \sqrt {t_k}/4

. Since

h(0)\ge g(0)

and

4\sigma _k'=\Lambda \sqrt {t_k}/2

\begin{equation*} \overline {B_{h(0)}(y,4\sigma _k')} \subset B_{g(0)}\left (x,\tfrac 34\Lambda \sqrt {t_k}\right ) \Subset B_{g(0)}(x,\Lambda \sqrt {t_k}). \end{equation*}

Moreover, pushing forward (5.10) and using

\eps _*=\eps _{\mathrm {DT}}/10

gives

\begin{equation*} (1-\eps _{\mathrm {DT}})h\le \widehat g_{s'} \le (1+\eps _{\mathrm {DT}})h\qquad (0\le s'\le t_k) \end{equation*}

on this ball. Every point of

\overline {B_{h(0)}(y,4\sigma _k')}

is of the form

F_{s'}(z)

with

z\in B_{g(0)}(x,2\Lambda \sqrt {t_k})

, for each

s'\le t_k

, and

B_{g(0)}(z,4\sigma _k')\subset B_{g(0)}(x,\tfrac 52\Lambda \sqrt {t_k})\subset B_{k,2}

. So Proposition 2.3(iii), applied at every such

z

with the radius

\sigma _k'

and

q=2

, gives

|(\nabla ^h)^2\widehat g_{s'}|_h\le K_2^{\mathrm {DT}}/s'

\overline {B_{h(0)}(y,4\sigma _k')}\times (0,t_k]

, by (5.9). Thus the hypotheses of Proposition 2.3(iv) hold on the closed cylinder over this ball, centred at

y

with radius

\sigma _k'

, and applying it with the exponent

m

and evaluating at

y

gives, for

0<s\le t_k

\begin{equation}\label {eq:lee-decay} |\widehat g_s-h(s)|_h \le C\left (\frac {64s}{\Lambda ^2t_k}\right )^m, \qquad |\nabla ^h\widehat g_s|_h^2 \le C\left (\frac {64}{\Lambda ^2t_k}\right )^{m+1}s^m.\tag{5.20} \end{equation}

Choose a dimensional

\delta _{\mathrm {met}}>0

so small that

\begin{equation*} |\widehat g_s-h(s)|_h\le \delta _{\mathrm {met}} \quad \hbox {implies}\quad (1-\eps _*/3)\widehat g_s\le h(s)\le (1+\eps _*/3)\widehat g_s. \end{equation*}

Choose

\Lambda

so that the first estimate in (5.20) is at most

\delta _{\mathrm {met}}

s=t_k

. Pulling back at

y=F_s(x)

gives the

1\pm \eps _*/3

estimate at

x

for

s\le t_k

. Since

x\in B_{k,5}

was arbitrary, (5.16), now with

\eps =\eps _*/3

and

C_{\mathrm {bar}}\mu \le \frac 23\eps _*

, gives exactly (5.10) on the new slab over

B_{k,5}

. This is the required estimate on the still smaller domain of

(\mathrm {Ind}_{k+1})

Moreover, the gauge identity $\partial _sF_s=-\mathcal V_s\circ F_s$ relating the two flows [Lee26], in which the DeTurck vector field satisfies $|\mathcal V_s|_h\le C|\nabla ^h\widehat g_s|_h$ , turns the second estimate in (5.20) into

\begin{equation*} |\partial _sF_s|_{h(s)} \le C t_k^{-(m+1)/2}s^{m/2},\qquad 0<s\le t_k, \end{equation*}

while Proposition 2.3(ii) gives

|\partial _sF_s|_{h(s)}\le Cs^{-1/2}

on the new slab. Measuring the trajectory in

h(t)

, splitting its integral at

t_k

, and using (5.7), we obtain, for

t_k\le t\le t_{k+1}

\begin{equation}\label {eq:disp-int} \begin {split} \int _0^{t_k}|\partial _sF_s|_{h(t)}\,ds &\le Ct^{a_*}t_k^{-(m+1)/2}\int _0^{t_k}s^{m/2-a_*}\,ds \le C\sqrt {t_k},\\ \int _{t_k}^{t}|\partial _sF_s|_{h(t)}\,ds &\le C\sqrt {t_k}. \end {split}\tag{5.21} \end{equation}

The first integral converges by the choice of

m

; on the second one the factor

(t/s)^{a_*}\le (1+\mu )^{a_*}

is absorbed into

C

. By (5.1), (5.21) and the shrinking-balls lemma for

h

\begin{equation}\label {eq:disp-conv} d_{g(0)}(x,F_t(x)) \le d_{h(0)}(x,F_t(x)) \le d_{h(t)}(x,F_t(x))+C(n,A_*)\sqrt t.\tag{5.22} \end{equation}

Increasing

\Lambda

once more makes the right-hand side of (5.22) at most

\Lambda \sqrt {t_{k+1}}/4

; this proves the first line of (5.11) on the new slab.

Covering. Fix

\begin{equation*} x\in B_{k+1,5}, \qquad R_k:=\Lambda \sqrt {t_{k+1}}, \end{equation*}

and put

\mathcal B=B_{g(0)}(x,2R_k)

. The definitions

\mathfrak r_{k+1}=\mathfrak r_k-6\Lambda \sqrt {t_k}

and

t_{k+1}\le (1+\mu )t_k

show that

\overline {\mathcal B}

lies both in

B_{k,3}

and in the region

B_{k,5}

where the new displacement estimate holds. Thus, for every

0\le t\le t_{k+1}

F_t

is an embedding on a neighbourhood of

\overline {\mathcal B}

and

\begin{equation*} d_{g(0)}(x,F_t(z)) \ge d_{g(0)}(x,z)-d_{g(0)}(z,F_t(z)) \ge \frac 74R_k>R_k \quad (z\in \partial \mathcal B), \end{equation*}

whereas

F_t(x)\in B_{g(0)}(x,R_k)

. Invariance of domain makes

F_t(\mathcal B)

open, and compactness of

\overline {\mathcal B}

gives

\partial F_t(\mathcal B)\subset F_t(\partial \mathcal B)

. The connected ball

B_{g(0)}(x,R_k)

does not meet this boundary and contains

F_t(x)

; hence it is contained in

F_t(\mathcal B)

. This proves the second line of (5.11) for the data

(\mathfrak r_{k+1},t_{k+1})

. Since

B_{k+1,0}\subset B_{k,5}\subset B_{k,3}

, properties (I) and (II) hold on

B_{k+1,0}\times [0,t_{k+1}]

as well, so

F

satisfies

(\mathrm {Ind}_{k+1})

.∎

5.3Proof of Lemma 5.1

It remains to start the induction, to run it, and to pass from the gauge estimates to the curvature comparison.

Proof of Lemma 5.1. The Initial Solution. Let $D'$ be a smooth domain with $D_{31}\Subset D'\Subset D_{32}$ . On $D_{64}$ the metrics $g(0)$ and $h(0)$ coincide, and so do all their time derivatives $\partial _t^kg(0)$ and $\partial _t^kh(0)$ , since both are Ricci flows of the same metric there and the time derivatives at $t=0$ are determined locally by the initial metric. Hence the identity map solves the Ricci–harmonic map heat flow to every order at $t=0$ , and the Dirichlet problem on $D'$ with initial and boundary value the identity has smooth compatible data. Short-time existence for this problem is classical [Ham75]; the argument there uses only the uniform parabolicity of the system and the compactness of $\overline {D'}$ and of a target neighbourhood of it, so it applies to time-dependent metrics. Let $F$ be the solution on $\overline {D'}\times [0,t_0]$ ; it is smooth up to $t=0$ with $F_0=\mathrm {id}$ , and its time derivative is bounded on this compact cylinder, so $d_{g(0)}(x,F_t(x))\le Ct$ and $dF_t$ is uniformly close to the identity for small $t$ . Decreasing $t_0$ therefore makes $F$ satisfy $(\mathrm {Ind}_0)$ for the data $(31,t_0)$ : (I) and (II) hold because a map $C^1$ -close to the identity on the compact set $\overline {D_{31}}$ is a diffeomorphism onto its image with the stated metric equivalence, and (III) follows from $Ct_0\le \Lambda \sqrt {t_0}/4$ and the covering argument at the end of the proof of Lemma 5.2. The time $t_0$ may depend on the initial geometry; the final time will not.

The Induction. Put $t_k:=(1+\mu )^kt_0$ , $\mathfrak r_0:=31$ and $\mathfrak r_{k+1}:=\mathfrak r_k-6\Lambda \sqrt {t_k}$ . Since the times are geometric,

\begin{equation*} \sum _{j\le k}\sqrt {t_j}\le C(\mu )\sqrt {t_k}. \end{equation*}

Choose

\tau _L=\tau _L(n,A_*,L)\le \tau '

so small that, whenever

t_k\le (1+\mu )\tau _L

, the total decrease

6\Lambda \sum _{j\le k}\sqrt {t_j}

of the radius is less than

7

and

2\Lambda \sqrt {t_k}<4

; then

\mathfrak r_{k+1}>24

and

D_8\subset B_{k+1,5}

. Decrease

t_0

below

\tau _L

. Repeated application of Lemma 5.2 gives

(\mathrm {Ind}_k)

for every

k\ge 1

with

t_{k-1}<\min \{T,\tau _L\}

and

t_k<T

: at each step the lemma applies, since

t_{k-1}<\tau _L\le \tau '

and

\mathfrak r_k>24

. If instead

t_k\ge T

, the last step with

t_k

replaced by any

t'\in (t_{k-1},T)

gives

(\mathrm {Ind}_k)

for the data

(\mathfrak r_k,t')

. Hence, for every

t<\min \{T,\tau _L\}

, the flow

F

exists on

D_{24}\times [0,t]

and satisfies (I)–(III) for the data of the slab containing

t

(31,t_0)

t\le t_0

, and, if

t_{k-1}<t\le t_k

, either

(\mathfrak r_k,t_k)

(\mathfrak r_k,t')

with

t\le t'<T

. In the second case the final time of the data is at most

(1+\mu )t

; in every case it is at most

(1+\mu )\tau _L

, so the choice of

\tau _L

applies to it.

Conclusion. By the choice of $\tau _L$ , the ball $D_8$ lies in the region $B_{k,5}$ where (III) holds. Restricting to $D_8$ gives $F_t(D_8)\subset D_{16}$ by the displacement estimate, and the second line of (5.11) with centres in $D_4$ gives $D_4\subset F_t(D_8)$ . For $t>t_0$ the final time of the data is at most $(1+\mu )t$ , so the first line of (5.11) gives the stated $C_{\mathrm {disp}}\sqrt t$ bound. For $t\le t_0$ , the estimate (5.20), which was derived from $(\mathrm {Ind}_k)$ alone, applied to the data $(31,t_0)$ , and (5.7) give, for $x\in D_8$ ,

\begin{equation*} \int _0^t|\partial _sF_s(x)|_{h(t)}\,ds \le C t_0^{-(m+1)/2}t^{(m+2)/2}\le C\sqrt t. \end{equation*}

The shrinking-balls lemma for

h

and (5.1) then give the same

g(0)

-displacement bound. This proves (5.3)–(5.4) for a time independent of the initial local curvature.

The Ricci–DeTurck Estimates. Let $t<\min \{T,\tau _L\}$ , with slab data $(\mathfrak r_k,t_k)$ . Since $2\Lambda \sqrt {t_k}<4$ , the second line of (5.11) with centres in $D_8$ gives $D_8\subset F_t(D_{16})$ , and $B_{g(0)}(z,4)\subset D_{24}\subset B_{k,0}$ for $z\in D_{16}$ . Hence Proposition 2.3(iii), applied at every $z\in D_{16}$ with $r=1$ and $q=4$ , gives

\begin{equation*} |(\nabla ^h)^j\widehat g|_h\le K_4^{\mathrm {DT}}t^{-j/2} \qquad (1\le j\le 4)\quad \hbox {on }D_8, \end{equation*}

after reducing

\tau _L

below

S_4(n,A_*,\Lambda _1,4)

. On

D_8

the metric

\widehat g

is a Ricci–DeTurck flow relative to

h

with

\widehat g(0)=h(0)

, and pushing forward (5.10) from

D_{16}

gives the

\eps _{\mathrm {DT}}

-equivalence there. For

y\in D_2

the closed ball

\overline {B_{h(0)}(y,4)}

lies in

D_8

because

h(0)\ge g(0)

, so Proposition 2.3(iv), with

r=1

and exponent

2L

, gives (5.5) on

D_2

after a further reduction of

\tau _L

. By the shrinking-balls lemma for

h

, an

h(t)

-ball of fixed radius about each point of

D_1

stays inside

D_2

for small

t

. One-dimensional covariant interpolation along geodesic segments in that ball, using

|(\nabla ^h)^4H|\le Ct^{-2}

and followed by polarization, gives

\begin{equation*} t|(\nabla ^h)^2H|\le C_Lt^{L/2}. \end{equation*}

The antisymmetric part of

(\nabla ^h)^2H

is bounded by

|\Rm _h||H|

. Finally,

\begin{equation*} |\Rm _{\widehat g}-\Rm _h|_h \le C\bigl (|(\nabla ^h)^2H|_h+|\nabla ^hH|_h^2 +|\Rm _h|_h|H|_h\bigr ), \end{equation*}

which proves (5.6). Restoring

\ell

yields the displayed scale factors. All constants depend only on

n,A_*,L

.∎

6Completion of the Bootstrap

The comparison flow of Section 4 and the comparison lemma of Section 5 are now combined. At the centre of the ball where the two initial metrics agree, $t|\Rm |$ for the given flow exceeds the bound $\beta$ of the comparison flow only by terms that tend to zero with $t/\ell ^2$ , so it is at most $\alpha$ on a time interval proportional to $r^2$ .

Proof of Proposition 3.1. Let $\beta$ , $\eps _P$ and $\delta _I$ be the constants of Section 4.1, and let $c_I$ , $\delta _*$ and $\kappa _*$ be as in (4.5). By hypotheses (a), (c) and (d), Lemma 4.1 applies to $g(0)$ with $\eta =\delta _I$ . Hence (4.3) and (4.4) hold for $g(0)$ , with $\eta =\delta _I$ and

\begin{equation*} R:=c_Ir. \end{equation*}

The Ricci lower bound (3.2) and the local cutoff construction [CC96, Theorem 6.33], in the form used in [HW22, proof of Lemma 2.2], give a smooth function

\chi

with values in

[0,1]

, supported in

B_{g(0)}(p,3R/4)

and equal to one on

B_{g(0)}(p,R/3)

, that satisfies (4.6) with a constant

C_\chi =C_\chi (n)

. Lemma 4.2 therefore gives a domain

\Omega

and a complete metric

h_0=e^{2u}g(0)

on it, with constants

c_H

and

C_H

that depend only on

n

and

\beta

, and Lemma 4.3 gives the Ricci flow

h(t)

h_0

[0,T_H]

, where

T_H=(\eps _Pr_H)^2

and

r_H=c_HR

Put

\begin{equation*} \ell :=\frac {R}{512},\qquad N:=\Omega . \end{equation*}

Then

\begin{equation*} D_{64}=B_{g(0)}(p,R/8) \Subset B_{g(0)}(p,R/4). \end{equation*}

Since

u\ge 0

N

, the conformal factor is at least one and

h(0)\ge g(0)

as quadratic forms on all of

N

; and

u=0

B_{g(0)}(p,R/4)

, so

h(0)=g(0)

D_{64}

. This is (5.1). Moreover

D_{64}\Subset N\Subset B_{g(0)}(p,3R/4)\Subset U

By (3.1) and (4.12),

\begin{equation}\label {eq:boot-curv} \sup _{D_{64}}|\Rm _{g(t)}|+\sup _N|\Rm _{h(t)}|\le \frac {A+\beta }{t} \qquad \bigl (0<t<\min \{T,T_H\}\bigr ).\tag{6.1} \end{equation}

Fix

T'<T

and put

T_c=\min \{T',T_H\}

. Apply Lemma 5.1 to

g

h

, and

N

[0,T_c]

, with

A_*=A+\beta

and

L=4

; hypothesis (5.2) is (6.1). Put

\tau =t/\ell ^2

. When

\tau <C_{\mathrm {disp}}^{-2}

, (5.3) places

F_t(p)

D_1

. Diffeomorphism invariance of curvature, (4.12), and (5.5)–(5.6) give

\begin{equation}\label {eq:boot-main} t|\Rm _{g(t)}|(p) \le (1+C_{\mathrm {met}}\tau )\left (\beta +C_4\tau ^2\right ).\tag{6.2} \end{equation}

Here

1+C_{\mathrm {met}}\tau

is the norm-comparison factor between

\widehat g

and

h

, with

C_{\mathrm {met}}=C_{\mathrm {met}}(n,A,\alpha )

, and

C_4

is the constant

C_L

of Lemma 5.1 for

L=4

Choose $\tau _0=\tau _0(n,A,\alpha )>0$ so that the right-hand side of (6.2) is at most $\alpha$ whenever $\tau \le \tau _0$ , and also

\begin{equation}\label {eq:tau0} \tau _0\le \min \left \{1,\tau _4,512^2(\eps _Pc_H)^2, \tfrac 14C_{\mathrm {disp}}^{-2},(2C_{\mathrm {met}})^{-1}\right \},\tag{6.3} \end{equation}

where

\tau _4

is the constant

\tau _L

of Lemma 5.1 for

L=4

. This choice depends only on

n,A,\alpha

. Setting

\begin{equation*} \Theta :=\frac {\tau _0c_I^2}{512^2} \end{equation*}

and using the

T_H

restriction in (6.3), the comparison estimate holds for every

t<\min \{T',\Theta r^2\}

. Letting

T'\uparrow T

proves (3.4). For

t\ge \Theta r^2

, the assumed bound gives

\begin{equation*} |\Rm |\le \frac At\le \frac {A}{\Theta }\,r^{-2}, \end{equation*}

which proves (3.5).∎

7Complete Noncompact Initial Manifolds

We now remove compactness of the initial manifold and prove Theorem 1.3. Throughout this section $(M^n,g_0)$ is complete and noncompact, $\Ric _{g_0}\ge -(n-1)g_0$ , and every point of $M$ is a $(\delta _{\mathrm {nc}},\rho )$ -LRRP, where $\delta _{\mathrm {nc}}$ is fixed in (7.1). We choose conformal completions whose initial metrics agree with $g_0$ on an exhaustion of $M$ . Uniform estimates for their Ricci flows on one common time interval give a locally smooth diagonal limit, and the agreement of the initial metrics with $g_0$ on the exhaustion identifies the initial value of that limit with $g_0$ .

7.1Constants

The conformal change costs a factor close to one in the isoperimetric constant, so the collar estimate needs slightly more room than was used in Section 4, and the constants are fixed accordingly. Let $\beta$ , $\eps _P$ , $\theta$ and $\delta _I$ be as in Section 4.1, and let $w_\theta$ be the conformal profile of Lemma 4.2. Put $\eta _0:=\delta _I/2$ , and let

\begin{equation*} c_0:=c(\eta _0),\qquad \delta _\flat :=\delta (\eta _0),\qquad \kappa _\flat :=\kappa (\eta _0) \end{equation*}

be the constants of Lemma 4.1 for $\eta =\eta _0$ . The constant of Theorem 1.3 is

\begin{equation}\label {eq:delta-nc} \delta _{\mathrm {nc}}:=\min \{\delta _0,\delta _\flat /2\}.\tag{7.1} \end{equation}

All these constants depend only on $n$ and $\alpha$ .

7.2The Conformal Exhaustion

The first lemma constructs the exhaustion and, for every $j$ and every point of $\Omega _j$ , a comparison flow on the local cover. The conformal change pushes the artificial boundary to infinity, while isoperimetric control on the local cover survives at the scale $e^{-u_j(x)}r_*$ of the original metric, which becomes a fixed scale for the conformal metric. This scale tends to zero towards $\partial \Omega _j$ , so the construction uses the LRRP condition at all scales below $\rho$ .

Lemma 7.1 (Uniform Conformal Exhaustion). There are relatively compact domains $\Omega _j\Subset M$ with smooth boundary, compact sets $E_j\subset \Omega _j$ whose union is $M$ , and smooth functions $u_j\ge 0$ on $\Omega _j$ such that $h_{j,0}:=e^{2u_j}g_0$ is a complete metric of bounded curvature on $\Omega _j$ and

\begin{equation}\label {eq:exhaustion} u_j=0\ \hbox {on }E_j,\qquad \overline {\Omega _j}\subset \Int E_{j+1}.\tag{7.2} \end{equation}

Moreover, there are

r_E,T_E>0

, depending only on

n,\rho ,\alpha

, with the following property. For every

j

and every

x\in \Omega _j

, let

\widetilde h_{j,0}:=\pi _x^*h_{j,0}

on the part of the local cover

\tB _{g_0}(x,\rho )

that lies over

\Omega _j

. There are an open set

N_{j,x}

in this part and a complete bounded-curvature Ricci flow

h^\sharp _{j,x}(t)

N_{j,x}\times [0,T_E]

such that, with

\ell _E:=r_E/512

\begin{equation}\label {eq:selected-E} \begin {split} &\overline {B_{\widetilde h_{j,0}}(\tilde x,64\ell _E)} \Subset N_{j,x},\\ &h^\sharp _{j,x}(0)\ge \widetilde h_{j,0}\quad \hbox {on }N_{j,x}, \qquad h^\sharp _{j,x}(0)=\widetilde h_{j,0} \quad \hbox {on } B_{\widetilde h_{j,0}}(\tilde x,128\ell _E),\\ &|\Rm _{h^\sharp _{j,x}(t)}|\le \frac {\beta }{t} \quad (0<t\le T_E). \end {split}\tag{7.3} \end{equation}

Figure 2. The conformal exhaustion of Lemma 7.1 (schematic, not to scale). Panels (a)–(c) show the
construction along a ray from the base point $o$ . (a) The regions. The metric $h_{j,0}=e^{2u_j}g_0$ equals $g_0$ where $u_j=0$ , in particular
on $E_j$ ; the boundary of $\Omega _j$ is at infinite distance for it, and $\overline {\Omega _j}$ lies in the interior of the next compact set $E_{j+1}$ .
(b) The cutoff $\chi _j$ , the regular value $a_j$ that defines $\Omega _j$ as a superlevel set, and the defining function $\varphi _j$ . (c) The
conformal exponent $u_j=w_\theta \circ \varphi _j$ , which vanishes where $\varphi _j\le 1-\frac 32\theta $ and tends to infinity at $\partial \Omega _j$ , and the function $e^{-u_j}$ , which is
Lipschitz and vanishes there. (d) The local cover at a point $x\in \Omega _j$ , with the metric $\widetilde h_{j,0}$ ; balls without a subscript
are taken for this metric. The second completion takes place at the fixed scale $r_E=512\ell _E$ , inside the ball $B_{\pi _x^*g_0}(\tilde x,c_0s)$ of the
lifted original metric, where $s=e^{-u_j(x)}r_*$ ; the radius of this ball tends to zero as $x$ approaches $\partial \Omega _j$ . It produces the
domain $N_{j,x}$ and the complete metric $h^\sharp _{j,x}(0)$ that starts the comparison flow. — Figure 2. The conformal exhaustion of Lemma 7.1 (schematic, not to scale). Panels (a)–(c) show the construction along a ray from the base point $o$ . (a) The regions. The metric $h_{j,0}=e^{2u_j}g_0$ equals $g_0$ where $u_j=0$ , in particular on $E_j$ ; the boundary of $\Omega _j$ is at infinite distance for it, and $\overline {\Omega _j}$ lies in the interior of the next compact set $E_{j+1}$ . (b) The cutoff $\chi _j$ , the regular value $a_j$ that defines $\Omega _j$ as a superlevel set, and the defining function $\varphi _j$ . (c) The conformal exponent $u_j=w_\theta \circ \varphi _j$ , which vanishes where $\varphi _j\le 1-\frac 32\theta$ and tends to infinity at $\partial \Omega _j$ , and the function $e^{-u_j}$ , which is Lipschitz and vanishes there. (d) The local cover at a point $x\in \Omega _j$ , with the metric $\widetilde h_{j,0}$ ; balls without a subscript are taken for this metric. The second completion takes place at the fixed scale $r_E=512\ell _E$ , inside the ball $B_{\pi _x^*g_0}(\tilde x,c_0s)$ of the lifted original metric, where $s=e^{-u_j(x)}r_*$ ; the radius of this ball tends to zero as $x$ approaches $\partial \Omega _j$ . It produces the domain $N_{j,x}$ and the complete metric $h^\sharp _{j,x}(0)$ that starts the comparison flow.

Proof.The Exhaustion. Fix $o\in M$ and $j\ge 1$ , and put $E_j:=\overline {B_{g_0}(o,2j)}$ . We first construct a cutoff at unit scale, as in [HW22, proof of Lemma 2.2]. Let $\{x_k\}$ be a maximal subset of $E_j$ whose points are pairwise at distance at least $1/4$ ; it is finite because $E_j$ is compact. The balls $B_{g_0}(x_k,1/4)$ cover $E_j$ , and the balls $B_{g_0}(x_k,1/8)$ are disjoint. By the Bishop–Gromov inequality, every point of $M$ lies in at most $C(n)$ of the balls $B_{g_0}(x_k,1/2)$ . The Ricci lower bound gives smooth cutoffs $b_k:M\to [0,1]$ with

\begin{equation*} b_k=1\ \hbox {on }B_{g_0}(x_k,1/4),\qquad \operatorname {supp}b_k\subset B_{g_0}(x_k,1/2),\qquad |\nabla b_k|+|\Delta b_k|\le C(n) \end{equation*}

[CC96, Theorem 6.33]. Fix a smooth nondecreasing

\upsilon _1:[0,\infty )\to [0,1]

with

\upsilon _1(0)=0

and

\upsilon _1=1

[1,\infty )

, and put

\chi _j:=\upsilon _1\circ \sum _kb_k

. Then

\begin{equation*} \chi _j=1\ \hbox {on }E_j,\qquad \operatorname {supp}\chi _j\subset \overline {B_{g_0}(o,2j+\tfrac 12)}, \qquad |\nabla \chi _j|+|\Delta \chi _j|\le C(n). \end{equation*}

Choose a regular value

a_j\in (1/4,1/2)

\chi _j

, let

\Omega _j

be the component of

\{\chi _j>a_j\}

containing

E_j

, and put on

\Omega _j

, with the function

\upsilon _0

of the proof of Lemma 4.2,

\begin{equation*} \varphi _j:=\upsilon _0\circ \frac {\chi _j-a_j}{1-a_j},\qquad u_j:=w_\theta \circ \varphi _j,\qquad h_{j,0}:=e^{2u_j}g_0. \end{equation*}

Thus

\varphi _j

vanishes on

E_j

, tends to one at

\partial \Omega _j

, and satisfies

\begin{equation}\label {eq:phi-j} |\nabla \varphi _j|+|\Delta \varphi _j|\le C(n).\tag{7.4} \end{equation}

As in the proof of Lemma 4.2, the last inequality in (4.10) and (7.4) show that every curve leaving compact subsets of

\Omega _j

has infinite

h_{j,0}

-length, so

h_{j,0}

is complete, and

h_{j,0}

has bounded curvature because

g_0

and

\varphi _j

extend smoothly past the compact set

\overline {\Omega _j}

. Since

\begin{equation*} \overline {\Omega _j}\subset \overline {B_{g_0}(o,2j+\tfrac 12)}\subset \Int E_{j+1}, \end{equation*}

this gives (7.2). Figure 2 shows these regions and functions.

Uniform Bounds. By (4.9), (7.4), the conformal scalar-curvature formula, and the bound $\Scal _{g_0}\ge -n(n-1)$ , there is a constant $C_E=C_E(n,\alpha )$ such that

\begin{equation}\label {eq:CE} |\nabla ^{h_{j,0}}u_j|_{h_{j,0}}\le C_E,\qquad \Scal _{h_{j,0}}\ge -C_E.\tag{7.5} \end{equation}

Indeed the derivatives of

w_\theta

occur only in the combinations of (4.9), and only the Laplacian of

\varphi _j

, not its full Hessian, enters the scalar-curvature formula.

The Adaptive Scale. Fix $x\in \Omega _j$ and put

\begin{equation*} s:=e^{-u_j(x)}r_*, \end{equation*}

where

r_*=r_*(n,\rho ,\alpha )\in (0,1]

will be fixed below. From (7.5),

\begin{equation*} |\nabla ^{g_0}e^{-u_j}|_{g_0}=|\nabla ^{h_{j,0}}u_j|_{h_{j,0}} \le C_E. \end{equation*}

e^{-u_j}

C_E

-Lipschitz for

d_{g_0}

along curves in

\Omega _j

, and it extends continuously by zero to

\partial \Omega _j

. Hence

d_{g_0}(x,\partial \Omega _j)\ge e^{-u_j(x)}/C_E

, and for every

c\in (0,1]

with

cC_Er_*\le \frac 16

\begin{equation}\label {eq:osc} \overline {B_{g_0}(x,cs)}\Subset \Omega _j,\qquad \bigl |e^{u_j(x)-u_j}-1\bigr |\le cC_Er_*\ \hbox { on }B_{g_0}(x,cs).\tag{7.6} \end{equation}

Isoperimetry on the Cover. Require

\begin{equation*} r_*\le \min \{\rho /20,\sqrt {\kappa _\flat }/4,1/20\}. \end{equation*}

Since

s\le r_*

, the closed ball of radius

s/2

about

\tilde x

in the local cover is compact, by Lemma 2.1; the Ricci lower bound gives

\Ric _{\pi _x^*g_0}\ge -(n-1)\kappa _\flat s^{-2}\pi _x^*g_0

; and the LRRP condition at

x

gives

\begin{equation*} d_{GH}\bigl (B_{\pi _x^*g_0}(\tilde x,2s),B_{\R ^n}(0,2s)\bigr ) <2\delta _{\mathrm {nc}}s\le \delta _\flat s . \end{equation*}

So Lemma 4.1 applies to

\pi _x^*g_0

\tilde x

, with

\eta =\eta _0

and

r=s

: every regular domain

E\Subset B_{\pi _x^*g_0}(\tilde x,c_0s)

satisfies

\begin{equation}\label {eq:iso-lift} \bigl (\area _{\pi _x^*g_0}\partial E\bigr )^n \ge (1-\eta _0)n^n\omega _n \bigl (\vol _{\pi _x^*g_0}E\bigr )^{n-1}.\tag{7.7} \end{equation}

The Conformal Metric on the Cover. Put $\eps _E:=c_0C_Er_*$ and require $\eps _E\le \frac 16$ . By (7.6) with $c=c_0$ , the ball $B_{\pi _x^*g_0}(\tilde x,c_0s)$ lies over $\Omega _j$ , its closure is compact, and on it the lift of $u_j$ , still written $u_j$ , satisfies $|e^{u_j(x)-u_j}-1|\le \eps _E$ . Comparing boundary areas and volumes for $\pi _x^*g_0$ and $\widetilde h_{j,0}=e^{2u_j}\pi _x^*g_0$ , we get from (7.7), for every regular domain $E\Subset B_{\pi _x^*g_0}(\tilde x,c_0s)$ ,

\begin{equation}\label {eq:iso-conf} \bigl (\area _{\widetilde h_{j,0}}\partial E\bigr )^n \ge \Bigl (\frac {1-\eps _E}{1+\eps _E}\Bigr )^{n(n-1)}(1-\eta _0)n^n\omega _n \bigl (\vol _{\widetilde h_{j,0}}E\bigr )^{n-1}.\tag{7.8} \end{equation}

Since

\eta _0=\delta _I/2

, we may decrease

r_*

so that the constant on the right is at least

(1-\delta _I)n^n\omega _n

. Set

\begin{equation*} r_E:=\frac {c_0r_*}{2(1+\eps _E)}, \end{equation*}

and decrease

r_*

once more so that

C_Er_E^2\le 1

. A curve from

\tilde x

to the boundary of

B_{\pi _x^*g_0}(\tilde x,c_0s)

has length at least

c_0s

for

\pi _x^*g_0

, hence at least

e^{u_j(x)}(1+\eps _E)^{-1}c_0s=2r_E

for

\widetilde h_{j,0}

. Therefore

B_{\widetilde h_{j,0}}(\tilde x,2r_E)\subset B_{\pi _x^*g_0}(\tilde x,c_0s)

, the closed

\widetilde h_{j,0}

-ball of radius

r_E

about

\tilde x

is compact, and by (7.5)

\begin{equation}\label {eq:scal-E} \Scal _{\widetilde h_{j,0}}\ge -r_E^{-2} \quad \hbox {on }B_{\widetilde h_{j,0}}(\tilde x,r_E),\tag{7.9} \end{equation}

while by (7.8) every regular domain

E\Subset B_{\widetilde h_{j,0}}(\tilde x,r_E)

satisfies (4.4) for the metric

\widetilde h_{j,0}

, with

\eta =\delta _I

The Cutoff on the Cover. The local cutoff construction [CC96, Theorem 6.33] for the metric $\pi _x^*g_0$ , whose Ricci curvature is at least $-(n-1)$ , gives a smooth function $\tilde \chi$ with values in $[0,1]$ such that

\begin{equation*} \tilde \chi =1\ \hbox {on }B_{\pi _x^*g_0}(\tilde x,c_0s/5), \qquad \operatorname {supp}\tilde \chi \subset B_{\pi _x^*g_0}(\tilde x,c_0s/4), \qquad s|\nabla \tilde \chi |+s^2|\Delta \tilde \chi |\le C, \end{equation*}

where the derivatives are taken for

\pi _x^*g_0

and

C=C(n,\alpha )

. The conformal Laplacian identity

\begin{equation*} \Delta _{\widetilde h_{j,0}}\tilde \chi =e^{-2u_j}\bigl (\Delta _{\pi _x^*g_0}\tilde \chi +(n-2)\langle \nabla u_j,\nabla \tilde \chi \rangle _{\pi _x^*g_0}\bigr ), \end{equation*}

together with (7.5), (7.6), and

s=e^{-u_j(x)}r_*

, gives

\begin{equation}\label {eq:cut-bounds} r_E|\nabla ^{\widetilde h_{j,0}}\tilde \chi |_{\widetilde h_{j,0}} +r_E^2|\Delta _{\widetilde h_{j,0}}\tilde \chi |\le C_\chi ',\tag{7.10} \end{equation}

with

C_\chi '=C_\chi '(n,\alpha )

. By the two-sided bound for

e^{u_j}

B_{\pi _x^*g_0}(\tilde x,c_0s)

, the

\widetilde h_{j,0}

-distance from

\tilde x

to the complement of

B_{\pi _x^*g_0}(\tilde x,c_0s/5)

is at least

2r_E/5>r_E/3

, and every point of

B_{\pi _x^*g_0}(\tilde x,c_0s/4)

is at

\widetilde h_{j,0}

-distance at most

c_0r_*/(4(1-\eps _E))<3r_E/4

from

\tilde x

, because

\eps _E\le \frac 16

. Hence

\begin{equation}\label {eq:cut-incl} \tilde \chi =1\ \hbox {on }B_{\widetilde h_{j,0}}(\tilde x,r_E/3),\qquad \operatorname {supp}\tilde \chi \subset B_{\widetilde h_{j,0}}(\tilde x,3r_E/4).\tag{7.11} \end{equation}

The Comparison Flow. By (7.9), the isoperimetric inequality stated after it, (7.10) and (7.11), Lemma 4.2 applies to the metric $\widetilde h_{j,0}$ , the point $\tilde x$ and the radius $R=r_E$ , with $C_\chi =C_\chi '$ . It gives a domain $N_{j,x}$ with

\begin{equation*} B_{\widetilde h_{j,0}}(\tilde x,r_E/4)\Subset N_{j,x} \Subset B_{\widetilde h_{j,0}}(\tilde x,3r_E/4) \end{equation*}

and a complete metric of bounded curvature on

N_{j,x}

that dominates

\widetilde h_{j,0}

, agrees with it on

B_{\widetilde h_{j,0}}(\tilde x,r_E/4)

, and satisfies the hypotheses of Lemma 4.3 with

r_H=c_H'r_E

, where

c_H'=c_H'(n,\alpha )

is the constant

c_H

of Lemma 4.2 for

C_\chi '

. Panel (d) of Figure 2 shows these regions. Let

h^\sharp _{j,x}(t)

be the flow of this metric given by Lemma 4.3. It exists on

[0,T_E]

with

\begin{equation*} T_E:=(\eps _Pc_H'r_E)^2, \end{equation*}

and it satisfies the last line of (7.3). Since

128\ell _E=r_E/4

, the other two lines hold as well.∎

7.3Uniform Lifetime

The second lemma is the first-time argument of Section 3 for the flows of the exhaustion metrics, with the flows $h^\sharp _{j,x}$ of Lemma 7.1 as comparison flows.

Lemma 7.2 (Uniform Lifetime of the Exhaustion Flows). There is $T_E'=T_E'(n,\rho ,\alpha )>0$ such that, for every $j$ , the complete bounded-curvature Ricci flow $h_j(t)$ with $h_j(0)=h_{j,0}$ exists on $[0,T_E']$ and satisfies

\begin{equation}\label {eq:lifetime} \sup _{\Omega _j}|\Rm _{h_j(t)}|\le \frac {\alpha }{t} \quad (0<t\le T_E').\tag{7.12} \end{equation}

Proof.Shi’s theorem [Shi89] starts a maximal complete bounded-curvature flow from each $h_{j,0}$ ; its lifetime $T_{\max ,j}$ may depend on $j$ . Define

\begin{equation*} T_j:=\sup \left \{s<T_{\max ,j}: t\sup _{\Omega _j}|\Rm _{h_j(t)}|\le 1 \text { for }0<t\le s\right \}. \end{equation*}

This number is positive for each

j

Fix $x\in \Omega _j$ and lift $h_j(t)$ to the part of the local cover $\tB _{g_0}(x,\rho )$ that lies over $\Omega _j$ . On every interval $[0,T]$ with $T<\min \{T_j,T_E\}$ , the lifted flow and the comparison flow $h^\sharp _{j,x}(t)$ satisfy the hypotheses of Lemma 5.1 with $N=N_{j,x}$ , $p=\tilde x$ , $\ell =\ell _E$ and $A_*=1+\beta$ : the first line of (7.3) gives $D_{64}\Subset N$ , (5.1) is its second line, and (5.2) follows from the definition of $T_j$ and its third line. As in the proof of Proposition 3.1, with $\tau =t/\ell _E^2$ and $L=4$ , this gives

\begin{equation*} t|\Rm _{h_j(t)}|(x)\le (1+C_{\mathrm {met}}\tau )\left (\beta +C_4\tau ^2\right ) \end{equation*}

for

\tau <C_{\mathrm {disp}}^{-2}

. Choose

\tau _E=\tau _E(n,\alpha )>0

so that the right-hand side is at most

\alpha

for

\tau \le \tau _E

and

\tau _E\le \min \{1,\tau _4,\frac 14C_{\mathrm {disp}}^{-2},(2C_{\mathrm {met}})^{-1}\}

, and put

\begin{equation*} T_E':=\min \{T_E,\tau _E\ell _E^2\}. \end{equation*}

Then, for every

x\in \Omega _j

\begin{equation}\label {eq:lifetime-improved} t|\Rm _{h_j(t)}|(x)\le \alpha \quad \bigl (0<t<\min \{T_j,T_E'\}\bigr ).\tag{7.13} \end{equation}

Suppose

T_j\le T_E'

. If

T_j<T_{\max ,j}

, the strict margin in (7.13) extends the defining inequality of

T_j

past

T_j

; the required continuity of the curvature supremum follows from the global derivative estimates of Shi on every compact subinterval of positive time. If

T_j=T_{\max ,j}

, (7.13) bounds the curvature on the final half of the maximal interval, and the continuation theorem for complete bounded-curvature flows extends the flow. Both alternatives are impossible. Hence

T_j>T_E'

, and (7.12) follows.∎

7.4The Limit Flow

The third lemma passes to the limit. It follows the compactness argument of Simon and Topping [ST21, Section 8, proof of Theorem 1.7]. Their existence theorem is three-dimensional and comes with a Ricci lower bound for positive time, but the two local estimates that it invokes and its diagonal extraction need neither.

Lemma 7.3 (Exhaustion Limit). After passing to a subsequence, the flows $h_j(t)$ converge locally smoothly on $M\times [0,T_E']$ to a complete Ricci flow $g(t)$ with $g(0)=g_0$ and

\begin{equation*} \sup _M|\Rm _{g(t)}|\le \frac {\alpha }{t} \qquad (0<t\le T_E'). \end{equation*}

Proof.Fix $i\ge 1$ and a radius $r\in (0,1]$ with $\sup _{E_{i+2}}|\Rm _{g_0}|\le r^{-2}$ , for the compact sets $E_i=\overline {B_{g_0}(o,2i)}$ of the proof of Lemma 7.1. Then

\begin{equation*} B_{g_0}(y,r)\subset E_{i+2}\qquad (y\in E_{i+1}). \end{equation*}

For all sufficiently large

j

u_j=0

on a neighbourhood of

E_{i+2}

. Since

h_{j,0}\ge g_0

, the

h_{j,0}

-ball of radius

r

about a point

y\in E_{i+1}

is then contained in

B_{g_0}(y,r)

, so the curvature of

h_{j,0}

is at most

r^{-2}

on it. The flow

h_j(t)

is complete, so its balls are relatively compact in

\Omega _j

, and it satisfies (7.12) with

\alpha <1

. These are the hypotheses of [Che09, Theorem 3.1], in the form recorded as [ST21, Lemma 8.1], at every centre

y\in E_{i+1}

. Its conclusion at the centre is

\begin{equation}\label {eq:chen} \sup _{E_{i+1}\times [0,T_E']}|\Rm _{h_j(t)}|\le e^{C(n)}r^{-2}\tag{7.14} \end{equation}

for all large

j

. The initial curvature derivatives are uniformly bounded on

E_{i+2}

, because the initial metric there is exactly

g_0

. Since

E_{i+1}

is a compact neighbourhood of

E_i

, (7.14) and the local derivative estimate [ST21, Lemma 8.2] give uniform bounds for all curvature derivatives on

E_i\times [0,T_E']

. Working in fixed coordinate charts, as in [ST21, Section 8, paragraph following Lemma 8.2], Arzelà–Ascoli gives a smooth subsequential limit on

E_i\times [0,T_E']

. A diagonal argument over

i

produces a smooth Ricci flow

g(t)

on all of

M\times [0,T_E']

. Since the initial metrics are equal to

g_0

on every fixed compact set for all large

j

, the convergence includes

t=0

and

g(0)=g_0

. Estimate (7.12) passes to the limit.

It remains to check completeness. The shrinking-balls lemma [ST21, Lemma 2.2], applied to the complete flow $h_j(t)$ with (7.12) and with arbitrarily large radii, gives, for a constant $C=C(n)$ and all $x,y\in \Omega _j$ ,

\begin{equation}\label {eq:shrinking} d_{h_j(t)}(x,y) \ge d_{h_{j,0}}(x,y)-C\sqrt {\alpha t} \ge d_{g_0}(x,y)-C\sqrt {\alpha t}.\tag{7.15} \end{equation}

For any curve

\gamma

M

from

x

y

, local smooth convergence and (7.15) give

\begin{equation*} \operatorname {length}_{g(t)}(\gamma ) =\lim _{j\to \infty }\operatorname {length}_{h_j(t)}(\gamma ) \ge d_{g_0}(x,y)-C\sqrt {\alpha t}. \end{equation*}

Taking the infimum over such curves gives the same lower bound for

d_{g(t)}(x,y)

. Thus every

g(t)

-ball about a fixed base point is contained in a compact

g_0

-ball, and the Hopf–Rinow theorem implies that

(M,g(t))

is complete for every

t\in [0,T_E']

.∎

The comparison lemma also gives the uniqueness part of Theorem 1.3.

Corollary 7.4 (Complete Type-I Uniqueness). Let $g(t)$ and $h(t)$ be complete Ricci flows, smooth on $M\times [0,T]$ , with the same complete smooth initial metric $g_0$ . If $|\Rm _g|+|\Rm _h|\le A/t$ on $M\times (0,T]$ for some $A<\infty$ , then $g(t)=h(t)$ on $[0,T]$ .

Proof.Fix a point $p$ in a connected component. For all sufficiently large $R$ , apply Lemma 5.1 with $N=U=M$ , $\ell =R$ , $A_*=\max \{2A,1\}$ and $L=2$ ; each of the two curvature norms is at most $A/t$ . Completeness of $g_0$ gives the compact balls required there, and $T\le \tau _2R^2$ , so the conclusions hold for $0<t<T$ . The resulting maps $F_{R,t}$ are defined on $B_{g_0}(p,8R)$ , satisfy $F_{R,0}=\mathrm {id}$ , and obey

\begin{equation*} d_{g_0}(x,F_{R,t}(x))\le C\sqrt t,\qquad |(F_{R,t}^{-1})^*g(t)-h(t)|_{h(t)} \le C(t/R^2)^2 \quad \hbox {on }B_{g_0}(p,2R). \end{equation*}

The first estimate keeps the images of each compact source set in a fixed compact target set, independently of

R

, while (5.4) uniformly bounds

|dF_R|

. On a positive-time compact cylinder, Proposition 2.3(ii) and interior parabolic regularity give uniform derivatives of

F_R

. Diagonal extraction therefore gives a smooth Ricci–harmonic map heat flow

F

M\times (0,T)

, continuous at

t=0

with

F_0=\mathrm {id}

. For any fixed compact set, its images lie in

B_{g_0}(p,2R)

for large

R

. Pulling back the second estimate, with only a uniform factor from (5.4), and taking the limit gives

F_t^*h(t)=g(t)

. Thus

F_t

is a local isometry, so its covariant Hessian and tension vanish. The Ricci–harmonic equation gives

\partial _tF=0

; continuity at

t=0

forces

F_t=\mathrm {id}

and hence

g(t)=h(t)

for

t<T

, and at

t=T

by continuity. Apply the argument on each component.∎

Proof of Theorem 1.3. Set $T_{\mathrm {nc}}:=\min \{T_0,T_E'\}$ , and recall that $\delta _{\mathrm {nc}}\le \delta _0$ by (7.1). If $M$ is closed, existence is Theorem 1.2 and uniqueness is Hamilton’s uniqueness theorem for closed manifolds [Ham82]. If $M$ is noncompact, Lemmas 7.1 to 7.3 give the flow $g(t)$ on $[0,T_E']$ . If two complete flows with initial metric $g_0$ satisfy a bound $|\Rm |\le A/t$ , the sum of their curvature norms is at most $2A/t$ , and the two flows coincide by Corollary 7.4.∎

8Singular Nilpotent Fibrations

At one fixed positive time, Theorem 1.2 supplies bounded-curvature metrics. The distance distortion estimate of [HW22, Appendix A] identifies their limit bi-Hölder homeomorphically with the original one, and the theory of Fukaya and Cheeger–Fukaya–Gromov supplies the fibration that we transport across this homeomorphism.

8.1The Fibration Theorem for Bounded Curvature

We use the theorem of Fukaya [Fuk87], [Fuk89, Theorem 9-1] and Cheeger–Fukaya–Gromov [CFG92] in the form of [Ron26b, Theorem 1.13]. For a metric $\bar g$ , equip $\Fr (M)$ with its canonical metric, so that $O(n)$ acts isometrically and $\Fr (M)/O(n)=(M,d_{\bar g})$ .

Theorem 8.1 (Nilpotent Fibrations under Bounded Curvature). Let $(M_i,\bar g_i)$ be closed $n$ -manifolds with $|\sec _{\bar g_i}|\le 1$ converging in the Gromov–Hausdorff topology to a compact metric space $\bar X$ with $\diam \bar X>0$ and $\dim \bar X<n$ . After passing to a subsequence, $\bar X=Y/O(n)$ for a closed manifold $Y$ carrying a $C^1$ Riemannian metric with Hölder continuous first derivatives and an isometric $O(n)$ -action. For $i$ large there is an $O(n)$ -equivariant fibre bundle $\hat f_i:\Fr (M_i)\to Y$ whose fibre is a nilmanifold and whose structure group is affine, inducing a continuous map $\bar f_i$ in the diagram

\begin{equation}\label {eq:diagram} \begin {array}{ccc} \Fr (M_i) & \xrightarrow {\ \hat f_i\ } & Y\\[3pt] \Big \downarrow & & \Big \downarrow \rlap {$\scriptstyle /O(n)$}\\[3pt] M_i & \xrightarrow {\ \bar f_i\ } & \bar X=Y/O(n), \end {array}\tag{8.1} \end{equation}

Every fibre of

\bar f_i

is a positive-dimensional infranilmanifold. Moreover, for some

\eps _i\to 0

(i): both $\hat f_i:\Fr (M_i)\to Y$ and $\bar f_i:(M_i,d_{\bar g_i})\to \bar X$ are $\eps _i$ -Gromov–Hausdorff approximations;
(ii): $e^{-\eps _i}|\xi |\le |d\hat f_i(\xi )|\le e^{\eps _i}|\xi |$ for $\xi$ orthogonal to a fibre;
(iii): the fibres of $\hat f_i$ have second fundamental form at most $c(n)$ .

8.2The Precise Statement

The parameter $\vartheta$ below governs the Hölder exponents relating the two limits.

Theorem 8.2 (Singular Nilpotent Fibrations, Precise Form). Fix $n\ge 2$ , $\rho >0$ and $\vartheta \in (0,1)$ , and put

\begin{equation*} \alpha :=\frac {\vartheta }{2\sqrt n\,(n-1)},\qquad \delta _1:=\delta _0(n,\alpha ), \end{equation*}

where

\delta _0

is the constant of Theorem 1.2. There is

\bar t=\bar t(n,\rho ,\vartheta )\in (0,1]

with the following property. Let

(M_i,g_i)

be closed

n

-manifolds with

\Ric _{g_i}\ge -(n-1)g_i

and uniform

(\delta _1,\rho )

-LRRP, converging in the Gromov–Hausdorff topology to a compact metric space

X

of positive diameter, with

\vol _{g_i}(M_i)\to 0

. Let

g_i(t)

be the Ricci flow with

g_i(0)=g_i

, and put

\bar g_i:=(\alpha /\bar t)\,g_i(\bar t)

. Then every subsequence has a further subsequence along which the following hold.

(a): The sequence $(M_i,\bar g_i)$ satisfies the hypotheses of Theorem 8.1 for some limit $\bar X$ . Let $Y$ , $\hat f_i$ , and $\bar f_i:M_i\to \bar X=Y/O(n)$ be as in that theorem.
(b): There is a homeomorphism $\Phi :X\to \bar X$ , the limit of the identity maps $(M_i,d_{g_i})\to (M_i,d_{\bar g_i})$ , such that $\Phi$ and $\Phi ^{-1}$ are Hölder continuous, with exponents $1-\vartheta$ and $1/(1+\vartheta )$ and with constants depending only on $n,\rho ,\vartheta$ , and $\diam X$ .
(c): The maps
$\begin{equation*} f_i:=\Phi ^{-1}\circ \bar f_i:M_i\longrightarrow X \end{equation*}$
are continuous and surjective, every fibre of $f_i$ is an infranilmanifold of positive dimension, and $f_i$ is an $\eps _i'$ -Gromov–Hausdorff approximation from $(M_i,d_{g_i})$ to $X$ , with $\eps _i'\to 0$ .

Consequently, for every $\eps >0$ , all but finitely many $M_i$ of the original sequence carry a map $f_i=\Phi ^{-1}\circ \bar f_i$ of this kind that is an $\eps$ -Gromov–Hausdorff approximation from $(M_i,d_{g_i})$ to $X$ , with $\hat f_i$ satisfying (i)–(iii) of Theorem 8.1 for $\eps$ in place of $\eps _i$ . Here $Y$ , $\bar X$ , and $\Phi$ may depend on $i$ , and the constants in (b) do not.

Thus $f_i$ fits into (8.1), with $X$ identified with $Y/O(n)$ through $\Phi$ . The estimates of Theorem 8.1 use the canonical metric of $\bar g_i$ ; the $g_i$ - and $\bar g_i$ -frame bundles are $O(n)$ -equivariantly diffeomorphic via $b\mapsto b\,(b^*\bar g_i)^{-1/2}$ .

8.3Uniform Bi-Hölder Distortion

The Ricci flow enters only through this comparison at one fixed time.

Lemma 8.3 (Uniform Bi-Hölder Distortion). Let $n,\rho ,\vartheta ,\alpha ,\delta _1$ be as in Theorem 8.2. There are $\bar t,\bar r\in (0,1]$ and $\bar C\ge 1$ , depending only on $n,\rho ,\vartheta$ , with the following property. Let $(M^n,g_0)$ be closed with $\Ric _{g_0}\ge -(n-1)g_0$ and uniform $(\delta _1,\rho )$ -LRRP, let $g(t)$ be its Ricci flow, and put $\bar g:=(\alpha /\bar t)\,g(\bar t)$ , $d:=d_{g_0}$ , and $\bar d:=d_{\bar g}$ . Then $|\sec _{\bar g}|\le 1$ , $\vol _{\bar g}(M)\le \bar C\vol _{g_0}(M)$ , and for all $x,y\in M$

\begin{align} \bar d(x,y)&\le \bar C\,d(x,y)^{1-\vartheta } &&\text {if }d(x,y)<\bar r, \label {eq:dist-upper}\tag{8.2}\\ d(x,y)&\le \bar C\,\bar d(x,y)^{1/(1+\vartheta )} &&\text {if }\bar d(x,y)<\bar r, \label {eq:dist-lower}\tag{8.3}\\ \bar d(x,y)&\le \bar C\bigl (d(x,y)+1\bigr ). \label {eq:dist-coarse}\tag{8.4}\end{align}

Proof.Put $T:=\min \{T_0,1\}$ , where $T_0=T_0(n,\rho ,\alpha )$ is given by Theorem 1.2. Its curvature estimate implies

\begin{equation*} |\Ric _{g(t)}|_{g(t)}\le \frac {\sqrt n\,(n-1)\alpha }{t} =\frac {2(n-1)\alpha '}{t}\quad (0<t\le T), \qquad \alpha ':=\frac {\vartheta }{4(n-1)}<\frac 1{2(n-1)}. \end{equation*}

The scalar lower bound

\Scal _{g_0}\ge -n(n-1)

persists by the maximum principle, since

M

is closed, and Bishop–Gromov and Buser’s inequality [Bus82] give dimensional doubling and Poincaré constants on balls of radius at most

10

. Thus [HW22, Theorem A.1], applied with arbitrary base point and coefficient

\alpha '

, gives

C_D=C_D(n)\ge 1

and

T_D=T_D(n,T)\in (0,T]

such that, for

r_D:=\sqrt {T_D}

and

d_t:=d_{g(t)}

\begin{equation}\label {eq:hw-distortion} C_D^{-1}d_0(x,y)^{1+\vartheta }\le d_t(x,y) \le C_Dd_0(x,y)^{1-\vartheta } \qquad \bigl (d_0(x,y)<r_D,\ 0\le t\le T_D\bigr ).\tag{8.5} \end{equation}

Here

4(n-1)\alpha '=\vartheta

; we use the estimate obtained at the end of the proof of [HW22, Theorem A.1].

Set $\bar t:=T_D$ , so that $\bar d=(\alpha /\bar t)^{1/2}d_{\bar t}$ and $|\sec _{\bar g}|\le 1$ . After taking $\bar r\le r_D$ and enlarging $\bar C$ , the upper half of (8.5) gives (8.2). Subdividing a minimal $g_0$ -geodesic into $k\le 2d(x,y)/r_D+1$ arcs of length at most $r_D/2$ gives $d_{\bar t}(x,y)\le kC_Dr_D^{1-\vartheta }$ and hence (8.4), after enlarging $\bar C$ .

Put $c_D:=C_D^{-1}(r_D/2)^{1+\vartheta }$ . If $d_{\bar t}(x,y)<c_D$ but $d(x,y)\ge r_D$ , continuity of $d(x,\cdot )$ along a minimizing $g(\bar t)$ -geodesic gives a point $z$ with $d(x,z)=r_D/2$ , and (8.5) gives the contradiction $d_{\bar t}(x,y)\ge d_{\bar t}(x,z)\ge c_D$ . Thus the lower half of (8.5) applies to $x,y$ and yields (8.3) after shrinking $\bar r\le (\alpha /\bar t)^{1/2}c_D$ and enlarging $\bar C$ .

Finally, since $\Scal _{g(t)}\ge -n(n-1)$ ,

\begin{equation*} \frac {d}{dt}\vol _{g(t)}(M)=-\int _M\Scal _{g(t)}\,d\vol _{g(t)} \le n(n-1)\vol _{g(t)}(M). \end{equation*}

Thus

\vol _{\bar g}(M)\le (\alpha /\bar t)^{n/2}e^{n(n-1)}\vol _{g_0}(M)

, and all constants depend only on

n,\rho ,\vartheta

.∎

8.4Proofs of Theorems 1.4 and 8.2

We now identify the two limits and transport the fibrations of the smoothed metrics.

Proof of Theorem 8.2. Let $\bar t,\bar r,\bar C$ be the constants of Lemma 8.3. Write $d_i:=d_{g_i}$ and $\bar d_i:=d_{\bar g_i}$ , and fix $\eps _i$ -Gromov–Hausdorff approximations $\psi _i:(M_i,d_i)\to X$ with $\eps _i\to 0$ . We start from a given subsequence and pass to further subsequences without changing the notation.

The Smoothed Limit. By Lemma 8.3, $|\sec _{\bar g_i}|\le 1$ , and (8.4) bounds the diameter of $(M_i,\bar g_i)$ by $\bar C(\diam X+2)$ for $i$ large. By Gromov’s precompactness theorem, a subsequence of $(M_i,\bar d_i)$ converges to a compact metric space $\bar X$ . Since $\vol _{\bar g_i}(M_i)\le \bar C\vol _{g_i}(M_i)\to 0$ , the sequence collapses and $\dim \bar X\le n-1$ [CC97, Theorem 3.1]. Choose $x_i,y_i\in M_i$ with $d_i(x_i,y_i)\to \diam X$ . By (8.3),

\begin{equation*} \bar d_i(x_i,y_i)\ge \min \bigl \{\bar r,\,(d_i(x_i,y_i)/\bar C)^{1+\vartheta }\bigr \}, \end{equation*}

\diam \bar X>0

. Thus Theorem 8.1 applies and gives (a). In particular the maps

\bar f_i

are continuous, surjective

\eps _i

-Gromov–Hausdorff approximations from

(M_i,\bar d_i)

\bar X

, after enlarging

\eps _i

The Homeomorphism. For $z\in X$ choose $x_i(z)\in M_i$ with $d_X(\psi _i(x_i(z)),z)<\eps _i$ , so that

\begin{equation}\label {eq:lift-points} \bigl |d_i(x_i(z),x_i(z'))-d_X(z,z')\bigr |<3\eps _i \qquad (z,z'\in X).\tag{8.6} \end{equation}

On a countable dense set

X'\subset X

, compactness and a diagonal subsequence define

\begin{equation*} \Phi (z):=\lim _i\bar f_i(x_i(z)). \end{equation*}

Passing (8.2)–(8.3) and (8.6) to the limit gives

\begin{align} d_{\bar X}(\Phi (z),\Phi (z'))&\le \bar C\,d_X(z,z')^{1-\vartheta },\label {eq:Phi-upper}\tag{8.7}\\ d_X(z,z')&\le \bar C\,d_{\bar X}(\Phi (z),\Phi (z'))^{1/(1+\vartheta )}.\label {eq:Phi-lower}\tag{8.8}\end{align}

The first holds when $d_X(z,z')<\bar r$ , and the second when the corresponding $\bar X$ -distance is less than $\bar r$ . Thus $\Phi$ extends continuously to $X$ , both inequalities persist under these restrictions, and $\Phi$ is injective.

We claim that

\begin{equation}\label {eq:Phi-close} \sup _{x\in M_i}d_{\bar X}\bigl (\Phi (\psi _i(x)),\bar f_i(x)\bigr ) \longrightarrow 0.\tag{8.9} \end{equation}

Indeed, fix

0<\eps <\bar r/3

and a finite

\eps

-dense set

X'_\eps \subset X'

. For large

i

\eps _i<\eps

and the defining convergence is within

\eps

X'_\eps

. Given

x\in M_i

, choose

z\in X'_\eps

with

d_X(\psi _i(x),z)<\eps

. Then

d_i(x,x_i(z))<3\eps

, and (8.2), (8.7), and the triangle inequality bound the distance in (8.9) by

2\bar C(3\eps )^{1-\vartheta }+2\eps

. Letting

\eps \downarrow 0

proves the claim.

Since $\bar f_i$ is surjective, (8.9) makes the compact set $\Phi (X)$ dense in $\bar X$ . Thus $\Phi$ is a homeomorphism, with the global Hölder bounds in (b) following from (8.7)–(8.8) and the diameter bounds.

The Fibrations. Put $f_i:=\Phi ^{-1}\circ \bar f_i$ . It is continuous and surjective with the same positive-dimensional infranilmanifold fibres, and (8.9) and (8.8) give

\begin{equation*} \sup _{x\in M_i}d_X\bigl (\psi _i(x),f_i(x)\bigr )\longrightarrow 0. \end{equation*}

Hence

f_i

is an

\eps _i'

-Gromov–Hausdorff approximation, where

\eps _i'

\eps _i

plus twice this supremum, and its diagram is (8.1) composed with

\Phi ^{-1}

Finally, if the whole-sequence assertion failed for some $\eps >0$ , its bad subsequence would have a good further subsequence by the argument above, a contradiction.∎

Proof of Theorem 1.4. Apply Theorem 8.2 with $\vartheta =\frac 12$ , so that $\delta _1(n)=\delta _0\bigl (n,(4\sqrt n\,(n-1))^{-1}\bigr )$ . By its last assertion there are indices $i_1<i_2<\cdots$ such that every $M_i$ with $i\ge i_k$ carries a map $f_i$ of the kind described there that is a $\frac 1k$ -Gromov–Hausdorff approximation. For $i_k\le i<i_{k+1}$ choose such a map and put $\eps _i=\frac 1k$ . Properties (i) and (ii) are part (c) of Theorem 8.2, and (iii) is the diagram (8.1), in which the manifold $Y$ may depend on $i$ .∎

8.5A Consequence for Aspherical Manifolds

The homeomorphism $\Phi$ transfers the topological conclusion of [Ron26b, Conjecture 0.7] under the hypotheses of case (0.6.1) of [Ron26b, Conjecture 0.6].

Corollary 8.4 (Aspherical Limits). Let $(M_i,g_i)$ and $X$ be as in Theorem 1.4. If every $M_i$ is aspherical and its fundamental group is of type $A_s$ in the sense of [Ron26b, Definition 0.4], then $X$ is homeomorphic to an aspherical manifold.

Proof.Apply Theorem 8.2 with $\vartheta =\frac 12$ . The bounded-curvature sequence $(M_i,\bar g_i)$ collapses to $\bar X$ with positive diameter, so [Ron26b, Theorem A] makes $\bar X$ an aspherical manifold; now use $\Phi :X\to \bar X$ .∎

9A Counterexample under Rewinding Volume Control

This section proves Theorem 1.5 and Corollary 1.6 independently of Sections 3 to 8, using the properties of Nifa’s manifolds listed below.

9.1Nifa’s Sequence

After the subsequences in [Nif26, Theorem 1.3], write $(\Sigma _j,g_j,p_j)\to (\Sigma _\infty ,d_\infty ,p_\infty )$ for Nifa’s pointed sequence and its limit. In [Nif26] they are denoted $M_j$ and $Y$ . Write $\Cone (Z)$ for the cone over $Z$ and $S^1_{2\pi {\mathfrak q}}$ for the circle of length $2\pi {\mathfrak q}$ ; the parameter $\mathfrak q$ is denoted $q$ in [Nif26]. We use exactly the following cited properties.

(N1): The Sequence. Each $\Sigma _j$ is a closed four-manifold, and there are $c_{\mathrm {Ric}},v_0>0$ with
$\begin{equation*} \Ric _{g_j}\ge c_{\mathrm {Ric}}\,g_j,\qquad \vol _{g_j}B_{\Sigma _j}(p_j,1)\ge v_0 . \end{equation*}$
The space $\Sigma _\infty$ is compact, and the convergence is realized by isometric embeddings of all $\Sigma _j$ and of $\Sigma _\infty$ into one compact metric space, called the ambient space below, in which $\Sigma _j\to \Sigma _\infty$ in the Hausdorff distance and $p_j\to p_\infty$ [Nif26, Theorem 1.3, Lemma 9.3, proof of Proposition 2.3].
(N2): The Singular Point. The tangent cone of $\Sigma _\infty$ at $p_\infty$ is unique and equal to $\R ^2\times \Cone (S^1_{2\pi {\mathfrak q}})$ , where $0<{\mathfrak q}<2\cdot 10^{-6}$ . The inclusion of any neighbourhood of $p_\infty$ into $\Sigma _\infty$ induces a map on $H_2(\,\cdot \,;\Z )$ whose image has infinite rank; in particular $\Sigma _\infty$ is not locally contractible at $p_\infty$ [Nif26, Theorem 1.3 and (7.1)].
(N3): The Cores. There are compact Riemannian four-manifolds with boundary $W_i$ , $i\ge 1$ , each diffeomorphic to $\CP ^2$ minus an open ball, such that for all $j\ge i$ the manifold $\Sigma _j$ contains an isometric copy of $W_i$ as a smooth domain, and the copies of different $W_i$ in $\Sigma _j$ are disjoint [Nif26, (9.3)]. Let $Q_i:=W_i/\partial W_i$ , a closed topological four-manifold homeomorphic to $\CP ^2$ , with base point $*$ the image of $\partial W_i$ , and let $\varpi _{i,j}:\Sigma _j\to Q_i$ be the quotient map on $W_i$ and the constant map to $*$ on $\Sigma _j\setminus \Int W_i$ .
(N4): The Limit Maps. There are continuous maps $\iota _i:W_i\to \Sigma _\infty$ and $\varpi _i:\Sigma _\infty \to Q_i$ such that $\varpi _i\circ \iota _i$ is the quotient map, $\iota _i$ is injective on $\Int W_i$ , the inclusions $W_i\to \Sigma _j$ converge to $\iota _i$ uniformly in the ambient space as $j\to \infty$ , and $\varpi _{i,j}(y_j)\to \varpi _i(y)$ whenever $\Sigma _j\ni y_j\to y$ in the ambient space [Nif26, Proposition 2.3 and its proof, Lemma 2.5].
(N5): The Regular Part. The set $\Sigma _\infty ^\circ :=\Sigma _\infty \setminus \{p_\infty \}$ is a connected topological four-manifold. It carries a smooth structure and a smooth Riemannian metric $g_\infty$ such that every point of $\Sigma _\infty ^\circ$ has a neighbourhood on which $d_\infty$ is the distance of $g_\infty$ . Moreover there are a compact domain $\mathcal K\subset \Sigma _\infty ^\circ$ with nonempty interior and smooth embeddings $\phi _j:\mathcal K\to \Sigma _j$ such that $\phi _j^*g_j\to g_\infty$ in $C^\infty (\mathcal K)$ and $\phi _j$ converges to the inclusion of $\mathcal K$ uniformly in the ambient space [Nif26, Lemma 14.1 and its proof].

Two consequences of (N3)–(N4) will be used. For $i\ne k$ and $y\in \Sigma _\infty$ ,

\begin{equation}\label {eq:wedge} \varpi _i(y)=*\quad \hbox {or}\quad \varpi _k(y)=*.\tag{9.1} \end{equation}

Indeed, for $y_j\to y$ , one of the two disjoint cores is avoided along a subsequence, and the corresponding quotient converges to $*$ . Also

\begin{equation}\label {eq:Wi} \mathcal W_i:=\varpi _i^{-1}(Q_i\setminus \{*\})=\iota _i(\Int W_i).\tag{9.2} \end{equation}

The reverse inclusion follows because $\varpi _i\circ \iota _i$ is the quotient map. For the forward inclusion, an approximating sequence for a point with $\varpi _i(y)\ne *$ eventually lies in a compact subset of $\Int W_i$ ; the inverse is $\iota _i$ composed with the inverse of the quotient there. Thus $\varpi _i|_{\mathcal W_i}:\mathcal W_i\to Q_i\setminus \{*\}$ is a homeomorphism, and $\mathcal W_i$ is an open four-manifold, disjoint from $p_\infty$ by (N2).

The lower bound for the Reifenberg constant also uses the shape of $\Sigma _j$ near the base point. The next lemma extracts what is needed from the construction.

Lemma 9.1 (Simply Connected Neighbourhoods of the Base Point). There are numbers $s_m\downarrow 0$ and a constant $C_\Sigma \ge 1$ such that for all $j\ge m$ there is an open, connected and simply connected set $\mathcal S_{m,j}\subset \Sigma _j$ with

\begin{equation*} B_{\Sigma _j}(p_j,s_m/64)\subset \mathcal S_{m,j}\subset B_{\Sigma _j}(p_j,C_\Sigma s_m). \end{equation*}

Proof.We use the construction of [Nif26, Section 14], in which $p_j=p$ for all $j$ and $\Sigma _j$ is obtained from the unit sphere $S^4$ by replacing disjoint round balls with the cores $W_i$ of (N3). Let $d_{S^4}$ be the round distance, and let $s_m$ be the numbers of the proof of [Nif26, Lemma 14.1]. They satisfy $s_{m+1}=s_m/2$ , the round ball that $W_i$ replaces is centred at $d_{S^4}$ -distance $\frac 43s_i$ from $p$ , and the open $d_{S^4}$ -ball of radius $s_m$ about $p$ contains the round balls replaced by $W_i$ for $i>m$ and is disjoint from those replaced by $W_i$ for $i\le m$ [Nif26, Section 14 and the proof of Lemma 14.1]. Let $\mathcal S_{m,j}\subset \Sigma _j$ be the open set obtained from this ball by the replacements of index $i>m$ made in $\Sigma _j$ . It is connected and simply connected by van Kampen, since finitely many disjoint balls are replaced by copies of $\CP ^2$ minus a ball.

Let $\mathsf f_j$ be the function on $\Sigma _j$ given by [Nif26, Lemma 13.2], where it is denoted $\lambda _j$ . It vanishes at $p$ , it is $32$ -Lipschitz, it equals $\frac 43s_i$ on $W_i$ , it lies between one half and twice $d_{S^4}(p,\cdot )$ outside the $W_i$ , and $d_{\Sigma _j}(p,\cdot )\le C_*\mathsf f_j$ for a constant $C_*$ independent of $j$ [Nif26, Lemma 13.2, (13.5) and (13.6)]. Moreover $\mathsf f_j\le 2s_m$ on $\mathcal S_{m,j}$ [Nif26, (14.4)]. Outside $\mathcal S_{m,j}$ , a point lies either on some $W_i$ with $i\le m$ , where $\mathsf f_j=\frac 43s_i\ge \frac 43s_m$ , or outside the $W_i$ at $d_{S^4}$ -distance at least $s_m$ from $p$ ; in both cases $\mathsf f_j\ge s_m/2$ . Since $\mathsf f_j\le 32\,d_{\Sigma _j}(p,\cdot )$ , every point at distance less than $s_m/64$ from $p$ lies in $\mathcal S_{m,j}$ , and every point of $\mathcal S_{m,j}$ is at distance at most $2C_*s_m$ from $p$ . This proves the inclusions with $C_\Sigma =2C_*+1$ .∎

9.2The Products

Let $\lambda _j\downarrow 0$ , with $\lambda _j<2/3$ , and equip

\begin{equation*} P_j:=\Sigma _j\times (\R /\lambda _j\Z ) \end{equation*}

with the product metric; write $[z,t]$ for the image of $(z,t)\in \Sigma _j\times \R$ . The next proposition verifies for $P_j$ the hypotheses of the statements recalled in the introduction.

Proposition 9.2 (The Hypotheses). Put $\mathfrak d:=\pi \sqrt {3/c_{\mathrm {Ric}}}$ and $v_*:=2v_0/(3^5(\mathfrak d+1)^4)$ .

(a): $\Ric _{P_j}\ge 0$ .
(b): Every point of every $P_j$ is a $(1,v_*)$ -LRVP.
(c): $d_{GH}(P_j,\Sigma _j)\le \lambda _j/2$ and $\vol (P_j)\le \lambda _j\,\omega _4\mathfrak d^4$ . Hence $P_j\to \Sigma _\infty$ and $\vol (P_j)\to 0$ .

Proof.(a) The circle factor is flat and $\Ric _{g_j}>0$ .

(b) By Myers’ theorem and (N1), $\diam \Sigma _j\le \mathfrak d$ . For $z\in \Sigma _j$ the ball $B_{\Sigma _j}(z,\mathfrak d+1)$ contains $B_{\Sigma _j}(p_j,1)$ , and the Bishop–Gromov inequality for $\Ric _{g_j}\ge 0$ gives

\begin{equation}\label {eq:allpoint} \vol B_{\Sigma _j}(z,\tfrac 13)\ge \frac {\vol B_{\Sigma _j}(z,\mathfrak d+1)}{3^4(\mathfrak d+1)^4} \ge \frac {v_0}{3^4(\mathfrak d+1)^4}.\tag{9.3} \end{equation}

For

x=[z,0]

, let

B'

be the component over

B_{P_j}(x,1)

\Sigma _j\times \R

containing

(z,0)

. Since

\lambda _j<2/3

, the connected cylinder

B_{\Sigma _j}(z,\frac 13)\times \R

lies in

B'

, and its rectangle with

|t|<\frac 13

lies in the intrinsic unit ball about

(z,0)

. The local-isometric cover

\tB _{P_j}(x,1)\to B'

maps the unit ball about

\tilde x

onto that ball by path lifting. Writing

g

for the product metric, (9.3) gives

\begin{equation*} \vol B_{\pi _x^*g}(\tilde x,1)\ge \vol \bigl (B_{\Sigma _j}(z,\tfrac 13)\times (-\tfrac 13,\tfrac 13)\bigr ) =\frac 23\vol B_{\Sigma _j}(z,\tfrac 13)\ge v_* . \end{equation*}

(c) The projection $P_j\to \Sigma _j$ is onto and changes distances by at most the diameter $\lambda _j/2$ of the circle. By the Bishop inequality, $\vol (\Sigma _j)\le \omega _4\mathfrak d^4$ .∎

9.3No Continuous Approximation

The obstruction is topological and concerns only $(\Sigma _j,\Sigma _\infty )$ .

Theorem 9.3 (No Continuous Approximation, General Form). There are $\eps _0>0$ and $j_0$ such that for $j\ge j_0$

\begin{equation*} \dis f\ge \eps _0\qquad \hbox {for every continuous map }f:\Sigma _j\to \Sigma _\infty . \end{equation*}

The same holds, with the same

\eps _0

and

j_0

, for every continuous map

f:\Sigma _j\times (\R /\lambda \Z )\to \Sigma _\infty

and every

\lambda >0

All degrees below are modulo two. For a proper map between four-manifolds, write $\deg _2(f;y)$ for the image of the fundamental class around the compact preimage of $y$ in the target’s local homology $\Z _2$ . It is locally constant in $y$ , and on closed connected four-manifolds it is the multiplier on $H^4(\,\cdot \,;\Z _2)$ [Dol95, Chapter VIII, §4]. The degree over a point is detected by the following form of the Borsuk–Ulam theorem.

Lemma 9.4 (Borsuk). Let a continuous map from the closed ball $\overline {B_{\R ^4}(0,r)}$ to $\R ^4$ take distinct values at any two points whose distance is at least $r$ . Then the image of the centre is not in the image of the boundary sphere, and the degree of the map over the image of the centre is odd.

Proof.Denote the map by $b$ . For $|x|=r$ and $0\le t\le 1$ we have $|x-(-tx)|\ge r$ , so $b(x)\ne b(-tx)$ . Thus the normalized map below is always defined; at $t=0$ it also shows that $b(0)$ is missed by the boundary and gives the degree map over $b(0)$ :

x\longmapsto \frac {b(x)-b(-tx)}{|b(x)-b(-tx)|}.

t=1

this map is odd, so its degree, and hence the degree of

b

over

b(0)

, is odd.∎

With it, a continuous map of small distortion is shown to have degree one over the regular part of $\Sigma _\infty$ .

Lemma 9.5 (The Degree Is One). There are $\eps _0>0$ and $j_0$ such that for $j\ge j_0$ and every continuous $f:\Sigma _j\to \Sigma _\infty$ with $\dis f<\eps _0$ , the proper map

\begin{equation*} f:f^{-1}(\Sigma _\infty ^\circ )\longrightarrow \Sigma _\infty ^\circ \end{equation*}

has

\deg _2=1

Proof.The restricted map is proper because the preimage of a compact subset of $\Sigma _\infty ^\circ$ is closed in $\Sigma _j$ . Its target is a connected four-manifold by (N5), so it suffices to find one point over which the degree is one.

Fix two points $y_1\ne y_2$ in the interior of $\mathcal K$ and smooth coordinate charts of $\Sigma _\infty ^\circ$ centred at them, with domains in the interior of $\mathcal K$ . By (N5), as in the proof of [Nif26, Lemma 14.1], there are $0<c_1<c_2$ , $r_2>0$ and $j_1$ such that for $j\ge j_1$ the composition $\phi _j^{(1)}$ of the inverse of the chart at $y_1$ with $\phi _j$ is defined on $B_{\R ^4}(0,r_2)$ and satisfies

\begin{equation*} \sqrt {c_1}\,|x-y|\le d_{\Sigma _j}(\phi _j^{(1)}(x),\phi _j^{(1)}(y))\le \sqrt {c_2}\,|x-y|, \qquad B_{\Sigma _j}\bigl (\phi _j^{(1)}(0),\tfrac 12\sqrt {c_1}\,r_2\bigr ) \subset \phi _j^{(1)}\bigl (B_{\R ^4}(0,\tfrac 12r_2)\bigr ), \end{equation*}

and the same holds for the chart at

y_2

. Put

c_3:=d_\infty (y_1,y_2)>0

. By (N5) there is

j_2

with

d_{\Sigma _j}(\phi _j(y_1),\phi _j(y_2))\ge c_3/2

for

j\ge j_2

The set of points of $\Sigma _\infty$ at distance at least $c_3/8$ from $p_\infty$ is a compact subset of the manifold $\Sigma _\infty ^\circ$ . By the Lebesgue number lemma there is $r_1>0$ such that every ball $B_{\Sigma _\infty }(y,r_1)$ about a point $y$ of this set lies in an open subset of $\Sigma _\infty ^\circ$ homeomorphic to $\R ^4$ . Choose

\begin{equation*} 0<r<\frac {r_2}2\ \hbox { with }\ 2\sqrt {c_2}\,r\le r_1, \qquad \eps _0:=\min \Bigl \{\frac {c_3}8,\ \frac {\sqrt {c_1}\,r}2,\ \frac {r_1}2\Bigr \}, \qquad j_0:=\max \{j_1,j_2\}. \end{equation*}

Let $j\ge j_0$ and $\dis f<\eps _0$ . The images of $\phi _j(y_1)$ and $\phi _j(y_2)$ under $f$ are at distance more than $c_3/2-\eps _0\ge 3c_3/8$ , so they are not both within $c_3/8$ of $p_\infty$ . Say $y':=f(\phi _j(y_1))$ is at distance at least $c_3/8$ from $p_\infty$ , and let $V$ be an open set homeomorphic to $\R ^4$ that contains $B_{\Sigma _\infty }(y',r_1)$ . Consider $b:=f\circ \phi _j^{(1)}$ on the closed ball of radius $r$ . Its image lies in $B_{\Sigma _\infty }(y',\sqrt {c_2}\,r+\eps _0)\subset V$ . If $|x-y|\ge r$ , then $d_{\Sigma _j}(\phi _j^{(1)}(x),\phi _j^{(1)}(y))\ge \sqrt {c_1}\,r\ge 2\eps _0$ , so $b(x)\ne b(y)$ . By Lemma 9.4, applied in $V\cong \R ^4$ , the degree of $b$ over $y'$ is odd. Finally, every point of $f^{-1}(y')$ is within $\eps _0$ of $\phi _j(y_1)=\phi _j^{(1)}(0)$ in $\Sigma _j$ . Since $\eps _0<\frac 12\sqrt {c_1}\,r_2$ , it lies in $\phi _j^{(1)}(B_{\R ^4}(0,\frac 12r_2))$ , and its coordinate has norm less than $\eps _0/\sqrt {c_1}<r$ . So $f^{-1}(y')\subset \phi _j^{(1)}(B_{\R ^4}(0,r))$ , and $\deg _2(f;y')=1$ .∎

Proof of Theorem 9.3. Let $\eps _0,j_0$ be as in Lemma 9.5, let $j\ge j_0$ , and suppose that $f:\Sigma _j\to \Sigma _\infty$ is continuous with $\dis f<\eps _0$ . For $i\ge 1$ put

\begin{equation*} \zeta _i:=(\varpi _i\circ f)^*\bar \zeta _i\in H^2(\Sigma _j;\Z _2), \end{equation*}

where

\bar \zeta _i

generates

H^2(Q_i;\Z _2)

. Here

i

runs over all positive integers, not only over the cores present in

\Sigma _j

: the maps

\varpi _i

are defined on

\Sigma _\infty

Squares. Let $z\in Q_i\setminus \{*\}$ , and let $y\in \mathcal W_i$ be its preimage under $\varpi _i$ . By (9.2), $(\varpi _i\circ f)^{-1}(z)=f^{-1}(y)$ , and $\varpi _i$ is a homeomorphism near $y$ . So $\deg _2(\varpi _i\circ f;z)=\deg _2(f;y)=1$ by Lemma 9.5. Since $\bar \zeta _i^2$ generates $H^4(Q_i;\Z _2)$ ,

\begin{equation*} \zeta _i^2=(\varpi _i\circ f)^*(\bar \zeta _i^2)\ne 0 \quad \hbox {in }H^4(\Sigma _j;\Z _2)=\Z _2 . \end{equation*}

Products. For $i\ne k$ the map $(\varpi _i,\varpi _k):\Sigma _\infty \to Q_i\times Q_k$ takes values in the wedge $Q_i\vee Q_k$ , by (9.1). The class $\bar \zeta _i\times \bar \zeta _k$ restricts to zero on the wedge, because it vanishes on each of the two factors. Hence $\varpi _i^*\bar \zeta _i\cup \varpi _k^*\bar \zeta _k=0$ on $\Sigma _\infty$ , and $\zeta _i\zeta _k=0$ .

Count. If $\sum _ic_i\zeta _i=0$ is a finite relation with $c_i\in \Z _2$ , multiplication by $\zeta _k$ gives $c_k\zeta _k^2=0$ , so $c_k=0$ . The classes $\zeta _1,\zeta _2,\dots$ are therefore linearly independent in $H^2(\Sigma _j;\Z _2)$ , which is finite-dimensional. This is a contradiction.

For the product, restrict a continuous map $f:\Sigma _j\times (\R /\lambda \Z )\to \Sigma _\infty$ to a slice $\Sigma _j\times \{\mathrm {pt}\}$ . The slice carries the distance of $\Sigma _j$ , so the restriction has distortion at most $\dis f$ .∎

9.4The Limit Is Not an Orbit Space

The limit cannot be the compact-group quotient occurring in a Cheeger–Fukaya–Gromov fibration.

Proposition 9.6 (Orbit Spaces Are Locally Contractible). Let a compact Lie group $G$ act smoothly on a smooth manifold $M$ . Then $M/G$ is locally contractible. Consequently $\Sigma _\infty$ is not homeomorphic to $M/G$ for any such $M$ and $G$ .

Proof.By the slice theorem [Bre72, Chapter VI, Section 2], a neighbourhood of $[z]\in M/G$ is modelled on a neighbourhood of $0$ in $V/G_z$ . A $G_z$ -invariant ball in $V$ contracts equivariantly to $0$ , so its quotient contracts; such quotients form a neighbourhood basis. Apply (N2).∎

9.5The Reifenberg Constant: Upper Bound

We quantify the facts that a ball in the local cover of a product contains a line segment through its centre and obeys Bishop–Gromov comparison.

Lemma 9.7 (A Crude Correspondence). Let $(Z,d)$ be a metric space with a point $z_0$ such that $d(z_0,z)<s$ for all $z\in Z$ . Let $0<\eps \le 1$ , let $m_1,m_2\ge 1$ be integers, and suppose that

(a): $S^{n-1}$ is the union of $m_1$ sets of diameter at most $2-\eps$ ;
(b): there are $z_1,\dots ,z_{m_1}\in Z$ with $d(z_0,z_i)\le (1-\eps )s$ and $d(z_i,z_k)\ge \eps s$ for $i\ne k$ ;
(c): $Z$ is the union of $m_2$ sets of diameter at most $(2-\eps )s$ ;
(d): there are $y_1,\dots ,y_{m_2}\in \R ^n$ with $|y_k|\le (1-\eps )s$ and $|y_k-y_l|\ge \eps s$ for $k\ne l$ .

Then there is a correspondence between $Z$ and $B_{\R ^n}(0,s)$ that contains $(z_0,0)$ and has distortion at most $(2-\eps )s$ . Hence the pointed Gromov–Hausdorff distance between $(Z,z_0)$ and $(B_{\R ^n}(0,s),0)$ is at most $(1-\frac \eps 2)s$ .

Proof.Let $\mathcal C_1,\dots ,\mathcal C_{m_1}$ be the sets in (a) and $Z_1,\dots ,Z_{m_2}$ those in (c), made disjoint. Relate $y\in B_{\R ^n}(0,s)\setminus \{0\}$ to $z_i$ if $y/|y|\in \mathcal C_i$ , relate $z\in Z$ to $y_k$ if $z\in Z_k$ , and relate $z_0$ to $0$ . Both projections are onto. For two pairs from the first rule, equal indices give distortion at most $(2-\eps )s$ by radial convexity and (a), while unequal indices compare numbers in $[\eps s,2(1-\eps )s]$ and $[0,2s)$ . The second rule is symmetric. For one pair of each kind, both distances are less than $(2-\eps )s$ ; a pair with $(z_0,0)$ compares numbers in $[0,s)$ . Thus the distortion is at most $(2-\eps )s$ , and the pointed bound follows from Section 2.1.∎

The next lemma bounds the number of sets needed in hypothesis (c).

Lemma 9.8 (Covering by Radial Retraction). Let $(M^n,g)$ be a complete Riemannian manifold with $\Ric _g\ge 0$ , let $x\in M$ and $0<s<\rho$ , and let $0<\eps <2a$ with $3a+\frac 12(a-\frac \eps 2)\le 1$ . Then the ball $B_{\pi _x^*g}(\tilde x,s)$ in the local cover is the union of at most

\begin{equation*} \Bigl (\frac {4a}{a-\eps /2}+1\Bigr )^n \end{equation*}

sets of diameter less than

(2-\eps )s

Proof.Put $r:=(a-\frac \eps 2)s$ and write $d$ for the lifted distance. Radially retract $y$ along a minimizing geodesic, available by Lemma 2.1, to $y^\flat$ at distance $a\,d(\tilde x,y)$ from $\tilde x$ . Then

\begin{equation*} d(y,y')<2(1-a)s+d\bigl (y^\flat ,y'^\flat \bigr ). \end{equation*}

Let

z_1,\dots ,z_k

be a maximal

r

-separated subset of

B_{\pi _x^*g}(\tilde x,as)

. The balls of radius

r

about the

z_i

cover

B_{\pi _x^*g}(\tilde x,as)

, and their radial preimages have diameter less than

2(1-a)s+2r=(2-\eps )s

. The

r/2

-balls are disjoint. If

z_*

has the least such volume, their union lies in

B(z_*,2as+r/2)

; moreover

z_*

has depth at least

\rho -as\ge 2as+r/2

. Hence Lemma 2.1 (iii) gives

\begin{equation*} k\le \frac {\vol B_{\pi _x^*g}(z_*,2as+\frac r2)}{\vol B_{\pi _x^*g}(z_*,\frac r2)} \le \Bigl (\frac {4as}{r}+1\Bigr )^n .\qedhere \end{equation*}

∎

The two lemmas give the upper bound.

Proposition 9.9 (The Upper Bound). Let $(M^4,g)$ be a complete Riemannian manifold with $\Ric _g\ge 0$ , let $\lambda >0$ , and let $P=M\times (\R /\lambda \Z )$ with the product metric. For every $x\in P$ , every $\rho >0$ and every $0<s<\rho$ , the pointed Gromov–Hausdorff distance between the ball of radius $s$ about $\tilde x$ in the local cover at $x$ and $B_{\R ^5}(0,s)$ is at most $\frac {11}{12}s$ . In particular every point of $P$ is a $(\delta ,\rho )$ -LRRP for every $\delta >\frac {11}{12}$ and every $\rho >0$ .

Proof.We verify the hypotheses of Lemma 9.7 with $n=5$ , $\eps =\frac 16$ , $m_1=10$ , and $Z$ the ball of radius $s$ about $z_0=\tilde x$ in the local cover at $x$ . For (a), the ten sets $\{y\in S^4:\pm y_i=\max _k|y_k|\}$ cover $S^4$ and have diameter at most $\sqrt {16/5}<11/6$ : in the first set, $\langle y,y'\rangle \ge 2/5-1$ . For (b), the circle projection lifts to a $1$ -Lipschitz function $\varsigma$ on $\tB _P(x,\rho )$ with $\varsigma (\gamma (t))=t$ along the arclength lift $\gamma$ from $\tilde x$ , for $|t|<\rho$ . Thus the lifted-circle points $\gamma (ks/6)$ , $-5\le k\le 4$ , are pairwise $s/6$ -separated and lie within $5s/6$ of $\tilde x$ . For (c), Lemma 9.8 with $a=0.29$ gives $m_2\le 12\,646$ sets. Finally, the $16\,875$ points $(s/6)k$ with $k\in \Z ^5$ and $|k|\le 5$ verify (d), since they are $s/6$ -separated and lie in the ball of radius $5s/6$ . Thus Lemma 9.7 gives $(1-1/12)s$ .∎

9.6The Reifenberg Constant: Lower Bound

A quantitative Borsuk–Ulam theorem separates the tangent model $\R ^3\times \Cone (S^1_{2\pi {\mathfrak q}})$ from $\R ^5$ .

Lemma 9.10 (Local Covers of Products). Let $\Sigma$ be a closed Riemannian manifold, $\lambda >0$ , and $x=[z,0]\in \Sigma \times (\R /\lambda \Z )$ . Let $0<s\le \rho /4$ , and suppose that $B_\Sigma (z,2s)$ is contained in a connected and simply connected set $S\subset \Sigma$ of diameter less than $\rho /4$ . Then the ball of radius $s$ about $\tilde x$ in the cover $\tB (x,\rho )$ is isometric to the ball of radius $s$ about $(z,0)$ in $\Sigma \times \R$ with the product distance, by an isometry that takes $\tilde x$ to $(z,0)$ .

Proof.Let $B'$ be the component over $B(x,\rho )$ in $\Sigma \times \R$ containing $(z,0)$ . The local cover factors through a local isometry $\pi ':\tB (x,\rho )\to B'$ , and $B'$ contains the product $\rho$ -ball. Product geodesics between points of the $s$ -ball have length below $2s$ and remain in the $3s$ -ball, so the product and intrinsic distances agree there; the same argument from $(z,0)$ identifies the intrinsic and product balls of radius at most $2s$ . Moreover every loop in the $2s$ -ball lies in the simply connected set $S\times (-2s,2s)\subset B'$ and hence lifts closed. Path lifting therefore maps the $s$ -ball about $\tilde x$ onto the product $s$ -ball. Given two lifted points, join their images by paths of length tending to their product distance; the resulting loops with paths from the base point lie in the $2s$ -ball and lift closed, so the lifted distance is no larger. The reverse inequality follows from local isometry.∎

We use unpointed Gromov–Hausdorff distance below; it is no larger than the pointed distance in (1.1).

Proposition 9.11 (The Model). Let $0<{\mathfrak q}\le \frac 12$ , and let $\overline B_{\mathfrak q}$ be the closed unit ball of $\R ^3\times \Cone (S^1_{2\pi {\mathfrak q}})$ about a point of the singular axis, with the restricted distance. Put $c_{\mathfrak q}:=1-\cos \pi {\mathfrak q}$ , and let $\eps _{\mathfrak q}$ be the smallest positive root of

\begin{equation*} \bigl (\tfrac 32-\tfrac 52c_{\mathfrak q}\bigr )\eps ^2+10(1+c_{\mathfrak q})\eps -10=0 . \end{equation*}

Then the Gromov–Hausdorff distance between

\overline B_{\mathfrak q}

and the closed unit ball of

\R ^5

is at least

\delta _{\mathrm {low}}({\mathfrak q}):=\eps _{\mathfrak q}/2

. The function

\delta _{\mathrm {low}}

is decreasing, with

\begin{equation*} \delta _{\mathrm {low}}({\mathfrak q})>0.4413\ \ \hbox {for }{\mathfrak q}\le \tfrac 1{100},\qquad \delta _{\mathrm {low}}(\tfrac 14)>0.37,\qquad \delta _{\mathrm {low}}(\tfrac 12)>0.25 . \end{equation*}

Proof.Let $\mathcal R$ be a correspondence between the closed unit ball of $\R ^5$ and $\overline B_{\mathfrak q}$ with distortion at most $\eps$ ; we may assume $\eps <2$ . For $x\in S^4$ choose a point of $\overline B_{\mathfrak q}$ that is related to $x$ , and write it as $(y(x),c(x))$ , with $y(x)\in \R ^3$ and $c(x)$ in the cone at distance $\varrho (x)$ from the vertex. Points of $\overline B_{\mathfrak q}$ satisfy $|y|^2+\varrho ^2\le 1$ , and the distance $d$ of two of them is given by

\begin{equation*} d^2=|y-y'|^2+\varrho ^2+\varrho '^2-2\varrho \varrho '\cos \angle (c,c'), \qquad 0\le \angle (c,c')\le \pi {\mathfrak q}\le \tfrac \pi 2 , \end{equation*}

where

\angle (c,c')

is the angle between the directions of

c

and

c'

. Put

\Pi (x):=(y(x),\varrho (x))\in \R ^4

. Then, for

x,x'\in S^4

and the distance

d

of the two chosen points,

\begin{equation}\label {eq:Pi} d^2-2c_{\mathfrak q}\varrho (x)\varrho (x')\le |\Pi (x)-\Pi (x')|^2\le d^2 .\tag{9.4} \end{equation}

Antipodes. For $x$ and $-x$ the distance of the chosen points is at least $2-\eps$ . Since the cosine above is nonnegative and $|y-y'|^2\le 2|y|^2+2|y'|^2\le 4-2\varrho ^2-2\varrho '^2$ , we get $(2-\eps )^2\le 4-\varrho ^2-\varrho '^2$ , that is, $\varrho (x)^2+\varrho (-x)^2\le 4\eps -\eps ^2$ . By (9.4),

\begin{equation*} |\Pi (x)-\Pi (-x)|^2\ge (2-\eps )^2-c_{\mathfrak q}(4\eps -\eps ^2)=:R^2 . \end{equation*}

Jumps. If $R^2\le 0$ , the inequality $4\eps ^2\ge \frac 52R^2$ obtained below is trivial, so assume $R>0$ . The map

\begin{equation*} x\longmapsto \frac {\Pi (x)-\Pi (-x)}{|\Pi (x)-\Pi (-x)|} \end{equation*}

from

S^4

S^3

sends antipodes to antipodes. By the theorem of Dubins and Schwarz [DS81], in the form of [LMS23, Corollary 5.3], its modulus of discontinuity is at least

\arccos (-\frac 14)

: for every

\eps '>0

there are

x',x''\in S^4

, arbitrarily close to each other, whose images are at distance at least

\sqrt {5/2}-\eps '

. On the other hand, by (9.4) and the distortion bound,

\begin{equation*} |\Pi (x')-\Pi (x'')|\le |x'-x''|+\eps , \end{equation*}

and the same holds at the antipodes. So the numerator changes by at most

2|x'-x''|+2\eps

between

x'

and

x''

. Its norm is at least

R

, and the radial projection onto the sphere of radius

R

does not increase distances outside that sphere, so the images of

x'

and

x''

are at distance at most

(2|x'-x''|+2\eps )/R

. Hence

2\eps /R\ge \sqrt {5/2}

, that is,

4\eps ^2\ge \frac 52R^2

. Written out, the quadratic polynomial in the statement is nonnegative at

\eps

, and it is negative at

0

. So

\eps \ge \eps _{\mathfrak q}

, and the Gromov–Hausdorff distance is half the infimum of the distortions. For fixed

\eps \in (0,4)

the polynomial increases with

c_{\mathfrak q}

, so its smallest positive root decreases as

\mathfrak q

increases. The numerical values follow from the formula for the root.∎

Together with the construction, Proposition 9.11 gives the lower bound in part (v) of Theorem 1.5.

Proposition 9.12 (The Lower Bound). For every $\rho >0$ and every $\delta <\delta _{\mathrm {low}}({\mathfrak q})$ there is $j_1$ such that for $j\ge j_1$ the point $[p_j,0]\in P_j$ is not a $(\delta ,\rho )$ -LRRP. In particular, for no $\delta \le 0.44$ and no $\rho >0$ do all the $P_j$ have uniform $(\delta ,\rho )$ -LRRP.

Proof.By Lemma 9.1 choose $m$ with $2C_\Sigma s_m<\rho /4$ , and let $0<s\le s_m/128$ . For $j\ge m$ the ball $B_{\Sigma _j}(p_j,2s)$ lies in the connected and simply connected set $\mathcal S_{m,j}$ , whose diameter is less than $\rho /4$ . By Lemma 9.10 the ball of radius $s$ about $\tilde x_j$ in the local cover at $x_j:=[p_j,0]$ is isometric to the ball of radius $s$ about $(p_j,0)$ in $\Sigma _j\times \R$ . By (N1), as $j\to \infty$ these balls converge in the Gromov–Hausdorff distance to the ball of radius $s$ about $(p_\infty ,0)$ in $\Sigma _\infty \times \R$ . By (N2), as $s\to 0$ the latter balls, rescaled by $1/s$ , converge to $\overline B_{\mathfrak q}$ . Therefore, by Proposition 9.11,

\begin{equation*} \liminf _{s\to 0}\ \liminf _{j\to \infty }\ \frac 1s\,d_{GH}\bigl (B_{\pi _{x_j}^*g}(\tilde x_j,s),B_{\R ^5}(0,s)\bigr ) \ge \delta _{\mathrm {low}}({\mathfrak q}), \end{equation*}

where

g

is the metric of

P_j

and the distance may be taken unpointed. This proves the first assertion. The second follows because

{\mathfrak q}<\frac 1{100}

by (N2).∎

9.7Proofs of Theorem 1.5 and Corollary 1.6

It remains to collect the pieces.

Proof of Theorem 1.5. Parts (i) and (ii) are Proposition 9.2; the diameter of $\Sigma _\infty$ is positive because $\Sigma _\infty$ has a tangent cone of dimension four. Part (iii) is Theorem 9.3. Part (iv) is (N2) and Proposition 9.6. Part (v) is Proposition 9.9, applied to $M=\Sigma _j$ , and Proposition 9.12.∎

Proof of Corollary 1.6. By parts (i) and (ii) of Theorem 1.5, the $P_j$ are closed five-manifolds with $\Ric \ge 0$ and uniform $(1,v_*)$ -LRVP that collapse with vanishing volume to the compact, positive-diameter limit $\Sigma _\infty$ , which has dimension four. Thus they satisfy the common hypotheses of the LRVP statements recalled in the introduction. A singular nilpotent fibration in the sense of [Ron26b, Section 1.3] is, in particular, a continuous $\eps _j$ -Gromov–Hausdorff approximation with $\eps _j\to 0$ : the frame-bundle map in (8.1) is one, and the vertical maps are quotients by isometric actions. So each of the statements (a)–(d) would provide, for all large $j$ , a continuous $\eps _j$ -Gromov–Hausdorff approximation $P_j\to \Sigma _\infty$ with $\eps _j\to 0$ . This contradicts part (iii) of Theorem 1.5. The last clause of (a) follows from part (v).∎

Remark 9.13 (Other Dimensions). Replacing the circle by $(\R /\lambda _j\Z )^k$ , with $\lambda _j<2/(3\sqrt k)$ , and using the radius $1/(3\sqrt k)$ in place of $\frac 13$ in the proof of Proposition 9.2 gives uniform $(1,v)$ -LRVP for some $v=v(k,c_{\mathrm {Ric}},v_0)>0$ , and hence analogues of parts (i)–(iv) of Theorem 1.5 and of Corollary 1.6 in every dimension $n\ge 5$ ; the obstruction follows by restriction to a slice.

10Discussion

10.1Between the Two Thresholds

In dimension five, Theorems 1.4 and 1.5 leave open the range between $\delta _1(5)$ and $0.44$ . The upper estimate $11/12$ of Proposition 9.9 is automatic for the circle products, whereas Proposition 9.11 shows that the conical model cannot produce a counterexample below $1/4$ when its cone angle is at most $\pi$ . A smaller counterexample would need a larger angle or a different mechanism. It is also open whether a positive threshold can be chosen independently of dimension, or whether $\Sigma _\infty$ can arise from another Ricci-lower-bounded sequence with uniform LRVP that does admit continuous approximations.

10.2Neighbouring Cases

The example of [Ron26b, Example 4.2] has no singular nilpotent fibration but does admit continuous approximations, so it does not decide parts (b)–(c) of Corollary 1.6: use the residual-circle orbit map and collapse the two small two-spheres over the poles to the vertices. The sectional-curvature case (0.6.2) of [Ron26b, Conjecture 0.6], with $\sec \ge -1$ , also remains open: such limits are locally contractible Alexandrov spaces [Kap06], unlike $\Sigma _\infty$ , while sectional lower bounds and LRVP do not imply small-constant LRRP, as shown by products of a shrinking circle with smooth convex surfaces converging to a surface with a conical point.

Acknowledgments and Tool Disclosure. The author used Anthropic’s Claude and OpenAI’s Codex for drafting and restructuring the exposition, for checking the arguments against the cited sources, for the numerical evaluation of the constants in Section 9, and for the preparation of the figures. The author takes full responsibility for the mathematical claims and the exposition.

ALocal Ricci–Harmonic Comparison Estimates

This appendix proves parts (i)–(iii) of Proposition 2.3. Lemma A.1 constructs the map heat flow on a conformal completion of the source; its existence time depends only on $n$ and $\Lambda _0$ , and the bounds of the construction that depend on the geometry enter only through the short-time existence lemma of [BK22]. Lemmas A.2 and A.3 give the interior derivative bounds.

A.1Interior Existence and Uniform Continuation

The flow is first constructed on a conformal completion of the source domain, and a comparison for its energy density makes the existence time uniform.

Lemma A.1 (Interior Existence). Under the hypotheses of Proposition 2.3(i), its conclusion holds with $2\Lambda _0$ in place of $10\Lambda _0$ .

Proof.Rescale $r=1$ . We use a fixed conformal completion of the source, as in [Hoc16; Lee26], and verify the bounds needed for a moving source. On a component with empty boundary take $G=g$ ; compactness gives all the qualitative bounds below. The following construction is for components with boundary.

Put $d(x)=\operatorname {dist}_{g(0)}(x,\partial U)$ , with paths in $\overline U$ . For $0<s\le 1/320$ and $d(x)>2s$ , let $\mathsf k_s$ be a normalized smooth radial kernel on $|v|<s$ in $T_xU$ and set

\begin{equation*} \mathcal M_sd(x)=\int _{T_xU}d(\exp _{x,g(0)}v)\,\mathsf k_s(v)\,dv, \end{equation*}

the Riemannian convolution of Greene and Wu [GW79]. These geodesics remain in

U

, and the curvature bound excludes conjugate points along them; since the integral is taken over the tangent space, no bound on the injectivity radius enters. We use the estimates

\begin{equation}\label {eq:map-smoothing-bounds} |\mathcal M_sd-d|\le s,\qquad |\nabla \mathcal M_sd|\le C_n,\qquad |\nabla ^2\mathcal M_sd|\le C_n/s.\tag{A.1} \end{equation}

The first is the Lipschitz bound on

d

. For the others, work in an orthonormal trivialization

(\partial _a)

that is parallel at

x

, and write

\begin{equation*} \mathcal E(x,v)=\exp _xv,\qquad \mathcal J_a=\partial _x\mathcal E(\partial _a),\qquad \widetilde {\mathcal J}_b=(\partial _v\mathcal E)^{-1}\mathcal J_b. \end{equation*}

Differentiating a smooth Lipschitz approximation of

d

twice in the centre produces

\begin{equation*} \nabla ^2d(\mathcal J_b,\mathcal J_a)+\langle \nabla d,\mathcal J_{ab}\rangle , \end{equation*}

where

\mathcal J_{ab}

is the covariant second centre variation. Since

\mathcal J_b=\partial _v\mathcal E(\widetilde {\mathcal J}_b)

, the Hessian term is a vertical derivative along

\widetilde {\mathcal J}_b

minus

\langle \nabla d,\nabla ^{\mathcal E}_{\widetilde {\mathcal J}_b}\mathcal J_a\rangle

, and integration by parts in

v

gives

\begin{equation*} (\nabla ^2\mathcal M_sd)_{ab} =-\int \operatorname {div}_v(\mathsf k_s\widetilde {\mathcal J}_b)\,\langle \nabla d,\mathcal J_a\rangle +\int \mathsf k_s\,\langle \nabla d,\mathcal J_{ab} -\nabla ^{\mathcal E}_{\widetilde {\mathcal J}_b}\mathcal J_a\rangle . \end{equation*}

The Jacobi equation and its first variation, together with the normalization

\begin{equation*} |\Rm _{g(0)}|+|\nabla \Rm _{g(0)}|\le 2, \end{equation*}

bound

\mathcal J_a

\widetilde {\mathcal J}_b

\mathcal J_{ab}

and

\nabla ^{\mathcal E}_{\widetilde {\mathcal J}_b}\mathcal J_a

, and

\|\operatorname {div}_v(\mathsf k_s\widetilde {\mathcal J}_b)\|_{L^1}\le C_n/s

. Since

|\nabla d|\le 1

, this proves (A.1), also in the limit for the distance function itself. Writing

\mathcal M_sd(x)=\int d(z)\,\mathsf K_s(x,z)\,dz

with a kernel that is smooth in

x

, because

\mathsf k_s

vanishes to all orders at

|v|=s

, shows that

\mathcal M_sd

is smooth; repeated centre differentiation and vertical integration by parts, with

\mathcal M_s1=1

, give

|\nabla ^q\mathcal M_sd|\le C_qs^{1-q}

, where

C_q

for

q\ge 3

also depends on the curvature derivatives of the compact smooth extension of the metric.

Take $\ell _j=2^{-j}/16$ and a fixed smooth nondecreasing function $\upsilon _2$ equal to zero below $2$ and one above $4$ . Define

\begin{equation*} \hat d(x)=\sum _{j\ge 0}\ell _j\,\upsilon _2\bigl (\mathcal M_{\ell _j/20}d(x)/\ell _j\bigr ). \end{equation*}

Each summand is defined where

d>\ell _j/10

and extended by zero; it already vanishes for

d<3\ell _j/2

, so

\hat d

is smooth. Only a bounded number of scales have nonzero derivatives at a point, and the smaller-scale tail is locally constant. Consequently

\begin{equation}\label {eq:map-boundary-radius} \hat d\asymp \min \{d,1\},\qquad |\nabla \hat d|+\hat d|\nabla ^2\hat d|\le C_n,\qquad |\nabla ^q\hat d|\le C_q\hat d^{1-q}.\tag{A.2} \end{equation}

Also

\hat d=1/8

U_1

. The constants

C_q

for

q\ge 3

may depend on the geometry.

The source assumptions give $|\partial _tg|+|\nabla ^g\partial _tg|\le C_n$ . For a dimensional $s_0>0$ , integration of the metric and connection equations gives $g(t)\asymp g(0)$ and $|\Gamma (g(t))-\Gamma (g(0))|_{g(0)}\le C_n$ for $t\le s_0$ . Thus (A.2) holds with $g(t)$ -derivatives as well. Fix a smooth nonnegative nonincreasing function $\upsilon _3$ with $\upsilon _3(z)=-\log z$ for $z\le 1/2$ and $\upsilon _3(z)=0$ for $z\ge 1$ , and set

\begin{equation*} u=\upsilon _3(64\hat d),\qquad G(t)=e^{2u}g(t). \end{equation*}

Then

G\ge g

G=g

U_1

, and

G(t)

is complete because

e^u\asymp d^{-1}

near

\partial U

. The conformal curvature formula gives

\begin{equation}\label {eq:map-completed-zero-order} |\Rm _G|_G+|\partial _tG|_G\le C_n \qquad (0\le t\le \min \{T,s_0\}).\tag{A.3} \end{equation}

Indeed

e^{-u}|du|+e^{-2u}|\nabla ^2u|\le C_n

and

|\partial _tG|_G=|\partial _tg|_g

All the following bounds are finite:

\begin{equation}\label {eq:map-completed-higher} \sup _{U\times [0,\min \{T,s_0\}]} \bigl (|(\nabla ^G)^j\Rm _G|_G+|(\nabla ^G)^j\partial _tG|_G\bigr )<\infty \qquad (j\ge 0).\tag{A.4} \end{equation}

For the first term, use derivatives of

u

through order

j+2

; the second also uses derivatives of

\partial _tg

through order

j

. All are available from the compact smooth extension and (A.2). Their boundary weights are respectively

e^{-(j+2)u}(1+\hat d^{-j-2})

and

e^{-ju}(1+\hat d^{-j})

, which are bounded. Unlike (A.3), these bounds depend on the geometry of the extension; they enter only through [BK22] below. The target Ricci flow has bounds of the same kind by hypothesis.

Now $f:(U,G(0))\to (N,h(0))$ is $\sqrt {\Lambda _0}$ -Lipschitz, and [BK22, Lemma A.8] gives a solution for the two complete evolving metric families on a short time interval, whose length may depend on (A.4). For $e_0=|dF|_{G,h}^2$ , take a $G$ -orthonormal frame $(v_i)$ . The Bochner identity for moving metrics is

\begin{equation*} \begin {split} (\partial _t-\Delta _G)e_0={}&-2|\nabla dF|^2 -\sum _{i,j}(\partial _tG+2\Ric _G)(v_i,v_j)\,h\bigl (dF(v_i),dF(v_j)\bigr )\\ &+\sum _i(\partial _th)\bigl (dF(v_i),dF(v_i)\bigr ) +2\sum _{i,j}\Rm _h\bigl (dF(v_i),dF(v_j),dF(v_j),dF(v_i)\bigr ), \end {split} \end{equation*}

where

\Rm _h(v_1,v_2,v_2,v_1)>0

when

v_1,v_2

span a positively curved plane. By (A.3) and

|\Rm _h|\le 1

(\partial _t-\Delta _G)e_0\le C_n(e_0+e_0^2)

. The maximum principle for bounded solutions on these complete families with bounded geometry, as in [BK22, proof of Lemma A.6], compares

e_0

with the solution of

y'=C_n(y+y^2)

y(0)=\Lambda _0

. Thus

e_0\le y<2\Lambda _0

up to

\begin{equation*} T_{\mathrm {ex}}=\min \left \{s_0,\frac 1{2C_n} \log \frac {2(1+\Lambda _0)}{1+2\Lambda _0}\right \}>0. \end{equation*}

Let $b=\min \{T,T_{\mathrm {ex}}\}$ . The bounds (A.4) and those of the target are uniform on $[0,b]$ , and $|dF_{t'}|_{G,h}^2\le 2\Lambda _0$ for $t'<b$ , so at every time $t'<b$ [BK22, Lemma A.8] supplies the same positive existence time for the initial map $F_{t'}$ . Restart within the preceding existence interval and concatenate: the two solutions agree at the joining time, and the equation determines all their time derivatives there from the spatial derivatives of the common map, so the one-sided jets match and the concatenation is smooth. The energy comparison on every finite concatenation continues to use the original initial time and gives $e_0\le y$ . Finitely many restarts cover $[0,b)$ . Restrict to $U_1$ , where $G=g$ , and restore the scale $r$ .∎

A.2Covariant Map Derivatives

The interior derivative bounds follow from a Shi-type induction on the covariant derivatives of the map.

Lemma A.2 (Local Map Derivatives). Let $g(t)$ be a Ricci flow on an open $n$ -manifold $M$ and $h(t)$ a complete Ricci flow on an $n$ -manifold $N$ . Suppose $B_{g(0)}(x_0,4r)\Subset M$ and

\sup _{B_{g(0)}(x_0,4r)}|\Rm _g|+\sup _N|\Rm _h|\le A/t.

F

is a smooth Ricci–harmonic map heat flow on this source cylinder with

|dF|^2\le \Lambda _0

, then, for every integer

q\ge 0

|\nabla ^jdF|^2\le K_qt^{-j}\quad (0\le j\le q)

B_{g(0)}(x_0,r)

for

0<t\le \min \{T,S_qr^2\}

, where

K_q,S_q

depend only on

n,A,\Lambda _0,q

Proof.Rescale $r=1$ and enlarge $A$ to $\max \{A,1\}$ . Local Shi estimates give, for a common smaller time and through order $q+1$ ,

\begin{equation*} |\nabla ^a\Rm _g|\le C_at^{-1-a/2}\quad \hbox {on }B_{g(0)}(x_0,3); \end{equation*}

by the shrinking-balls lemma [ST21, Lemma 2.2], the parabolic neighbourhoods of radius

c\sqrt t

used on

[t/2,t]

stay inside

B_{g(0)}(x_0,4)

. Complete Shi estimates give the same bounds for all target curvature derivatives.

Put $e_k=|\nabla ^kdF|^2$ , and let $\mathcal D_t$ denote target covariant differentiation along $F$ in time, leaving the source indices unchanged. For a tensor $\mathcal T$ with covariant source slots and one target slot,

\begin{equation}\label {eq:evolving-covariant-norm} \partial _t|\mathcal T|^2=2\langle \mathcal D_t\mathcal T,\mathcal T\rangle +2\sum _a\langle \Ric _g^{\sharp (a)}\mathcal T,\mathcal T\rangle -2\langle \Ric _h^\sharp \mathcal T,\mathcal T\rangle ,\tag{A.5} \end{equation}

the sum running over the source slots. The Ricci endomorphisms act on the indicated source slot and on the target slot, respectively; the target tensor is pulled back along

F

. Spatial metric compatibility therefore gives

\begin{equation}\label {eq:ek-evolution} (\partial _t-\Delta )e_k =2\langle (\mathcal D_t-\Delta )\nabla ^kdF,\nabla ^kdF\rangle -2e_{k+1} +\mathcal Q_k,\qquad |\mathcal Q_k|\le C_kt^{-1}e_k,\tag{A.6} \end{equation}

where

\mathcal Q_k

collects the Ricci terms of (A.5).

Commuting $\nabla$ through the tension field, and using that $\partial _t\Gamma (g)$ is a universal contraction of $g^{-1}$ and $\nabla \Ric _g$ , together with the target analogue, gives universal contractions of the forms

\begin{equation}\label {eq:commutator-types} \nabla ^a\Rm _g\otimes \nabla ^bdF,\quad a+b=k;\qquad (\widetilde \nabla ^a\Rm _h)\circ F\otimes \nabla ^{j_1}dF\otimes \cdots \otimes \nabla ^{j_m}dF, \quad a+\textstyle \sum _lj_l=k.\tag{A.7} \end{equation}

The number of factors depends only on

k

, and the factors

dF

are bounded. These relations follow inductively: a derivative either raises a curvature order or a jet index, with an extra undifferentiated

dF

when it differentiates target curvature. The moving-target connection terms have the same relations. Consequently, if

|\nabla ^jdF|\le Ct^{-j/2}

for

j<k

|(\mathcal D_t-\Delta )\nabla ^kdF|\le Ct^{-1}|\nabla ^kdF|+Ct^{-(k+2)/2}.

Terms containing

\nabla ^kdF

have curvature order zero and only

dF

as other jet factors; all other terms are controlled by the lower jets. For

k=0

the same bound follows directly using

|dF|^2\le \Lambda _0

Suppose the bounds hold through $e_k$ . Equations (A.6)–(A.7) and Young’s inequality imply

\begin{align} (\partial _t-\Delta )e_k &\le -2e_{k+1}+Ct^{-k-1},\label {eq:ek}\tag{A.8}\\ (\partial _t-\Delta )e_{k+1} &\le -2e_{k+2}+Ct^{-1}e_{k+1}+Ct^{-k-2}. \label {eq:ekplus}\tag{A.9}\end{align}

For the localization take $r_k=3-k/(q+1)$ . At the step from $k$ to $k+1$ , centre a cutoff at $z\in B_{g(0)}(x_0,r_{k+1})$ and put $s=(r_k-r_{k+1})/2$ . A nonincreasing distance cutoff

\Upsilon =\upsilon _4\bigl ((d_{g(t)}(\cdot ,z)+c(n)\sqrt {At})/s\bigr )

has compact support in

B_{g(0)}(z,s)\Subset B_{g(0)}(x_0,r_k)

, equals one at

z

for a common smaller time, and satisfies

|\nabla \Upsilon |^2\le C\Upsilon ,\qquad (\partial _t-\Delta )\Upsilon \le C\Upsilon .

Here

\upsilon _4

is a fixed smooth nonincreasing function equal to one below

1/2

and zero above

1

, with

|\upsilon _4'|^2\le C\upsilon _4

and

\upsilon _4''\ge -C\upsilon _4

. The shifted distance is a parabolic supersolution on the transition annulus, and by the shrinking-balls lemma the support stays in

B_{g(0)}(z,s)

. At cut points the calculation is understood in the barrier sense.

Consider $\Xi =\mathsf B t^ke_k+t^{k+1}\Upsilon e_{k+1}$ . At a positive-time interior maximum, (A.8)–(A.9) give

0\le (-2\mathsf B+C)t^ke_{k+1} -2t^{k+1}\Upsilon e_{k+2}+C(\mathsf B+1)t^{-1} -2t^{k+1}\langle \nabla \Upsilon ,\nabla e_{k+1}\rangle .

The last term is at most

t^{k+1}\Upsilon e_{k+2} +Ct^{k+1}(|\nabla \Upsilon |^2/\Upsilon )e_{k+1}

. For

t\le 1

and large

\mathsf B

, it follows that

t^{k+1}e_{k+1}\le C

at the maximum. On the lateral boundary

\Upsilon =0

and

t^ke_k

is already bounded. At

t=0

the second term vanishes; the first is bounded by

\mathsf B\Lambda _0

when

k=0

and vanishes otherwise, since

F

is smooth on the closed cylinder. Thus the maximum principle bounds

\Xi

, and

\Upsilon (z,t)=1

gives

e_{k+1}(z,t)\le Ct^{-k-1}

. The constants are uniform in

z

. Induction reaches

B_{g(0)}(x_0,r_q)\supset B_{g(0)}(x_0,1)

. Restoring

r

proves the statement.∎

A.3Ricci–DeTurck Derivatives

The derivative bounds for the Ricci–DeTurck flow are the map derivative bounds of Lemma A.2 in disguise: the gauge identity below expresses the covariant derivatives of $\widehat g$ through the Hessian of $F$ .

Lemma A.3 (DeTurck Derivatives from Map Derivatives). Let $g(t)$ , $h(t)$ , $x_0$ , $r$ , $\Lambda _0$ and $F$ be as in Lemma A.2, and suppose that every $F_t$ is a diffeomorphism onto its image with $(1-\eps )g(t)\le F_t^*h(t)\le (1+\eps )g(t)$ on $B_{g(0)}(x_0,4r)$ for some $\eps \le \frac 12$ . Then $\widehat g=(F_t^{-1})^*g$ satisfies, for every integer $q\ge 1$ ,

|(\nabla ^h)^j\widehat g|_h\le K_q^{\mathrm {DT}}t^{-j/2}\qquad (1\le j\le q)

F_t(B_{g(0)}(x_0,r))

for

0<t\le \min \{T,S_qr^2\}

, where

K_q^{\mathrm {DT}}

depends only on

n,A,\Lambda _0,q

and

S_q

is the constant of Lemma A.2.

Proof.Fix $t$ and write $\check h:=F_t^*h(t)$ , a metric on $B_{g(0)}(x_0,4r)$ for which $F_t$ is an isometry onto its image, and $g=F_t^*\widehat g$ . Pull-back commutes with covariant differentiation, so

F_t^*\bigl ((\nabla ^h)^j\widehat g\bigr )=(\nabla ^{\check h})^jg, \qquad \bigl |(\nabla ^h)^j\widehat g\bigr |_h\circ F_t =\bigl |(\nabla ^{\check h})^jg\bigr |_{\check h}.

The two connections on the source are related through the Hessian of

F_t

: since

dF_t

intertwines

\nabla ^{\check h}

with

\nabla ^h

, for vector fields

v_1,v_2

on the source,

\nabla ^{\check h}_{v_1}v_2-\nabla ^g_{v_1}v_2=dF_t^{-1}\bigl (\nabla dF_t(v_1,v_2)\bigr ), \qquad \nabla dF_t(v_1,v_2):=\nabla ^h_{dF_tv_1}(dF_tv_2)-dF_t(\nabla ^g_{v_1}v_2).

Because

\nabla ^gg=0

, this gives

(\nabla ^{\check h}_{v_1}g)(v_2,v_3) =-g\bigl (dF_t^{-1}\nabla dF_t(v_1,v_2),v_3\bigr )-g\bigl (v_2,dF_t^{-1}\nabla dF_t(v_1,v_3)\bigr ),

\nabla ^{\check h}g

is a universal contraction of

g

dF_t^{-1}

and

\nabla dF_t

. Differentiating again, with the connection

\nabla ^{\check h}

written as

\nabla ^g+dF_t^{-1}\nabla dF_t

and with

\nabla (dF_t^{-1})=-dF_t^{-1}\circ \nabla dF_t\circ dF_t^{-1}

, shows inductively that

(\nabla ^{\check h})^jg

is a sum of universal contractions of

g

dF_t^{-1}

and tensors

\nabla ^{i_1}dF_t,\dots ,\nabla ^{i_m}dF_t

with

i_1+\dots +i_m=j

and all

i_l\ge 1

: each differentiation raises the order of one factor by one or introduces a factor

\nabla dF_t

. The metric equivalence bounds

dF_t^{-1}

in operator norm by

(1-\eps )^{-1/2}

, hence in tensor norm by

\sqrt n(1-\eps )^{-1/2}

, and makes the norms taken with

g

and with

\check h

comparable, so Lemma A.2 gives, on

B_{g(0)}(x_0,r)

and for

0<t\le \min \{T,S_qr^2\}

\bigl |(\nabla ^{\check h})^jg\bigr |_{\check h} \le C\sum \prod _l|\nabla ^{i_l}dF_t| \le C\sum \prod _lK_q^{1/2}t^{-i_l/2} \le K_q^{\mathrm {DT}}t^{-j/2},

the sums running over the finitely many contractions. Composing with

F_t^{-1}

proves the lemma.∎

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How to cite

Shaosai Huang (2026). On the Collapsing of Manifolds with Local Rewinding Reifenberg Control. Working paper, version of October 2026. Kspectra Research. https://kspectra.ai/papers/collapsing-manifolds-local-rewinding-reifenberg/

@misc{huang2026collapsing,
  author = {Huang, Shaosai},
  title  = {{On the Collapsing of Manifolds with Local Rewinding Reifenberg Control}},
  year   = {2026},
  month  = oct,
  note   = {Working paper, version of October 2026},
  url    = {https://kspectra.ai/papers/collapsing-manifolds-local-rewinding-reifenberg/}
}

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