--- title: "On the Collapsing of Manifolds with Local Rewinding Reifenberg Control" authors: - name: "Shaosai Huang" affiliation: "Kspectra Research Inc., Toronto, Canada" date: "2026-10" status: "Working paper" url: https://kspectra.ai/papers/collapsing-manifolds-local-rewinding-reifenberg/ --- # On the Collapsing of Manifolds with Local Rewinding Reifenberg Control Shaosai Huang — Working paper, version of October 2026. Links: [Web page](https://kspectra.ai/papers/collapsing-manifolds-local-rewinding-reifenberg/) > Converted by Kspectra Research from the LaTeX of the posted version. Section, theorem, equation and reference numbers match the PDF. Formulas are LaTeX; the document's own macros are defined below. ## How to cite ```bibtex @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/} } ``` ## Macros The formulas use these definitions from the paper's preamble: ```latex \renewcommand{\topfraction}{0.85} \renewcommand{\bottomfraction}{0.6} \renewcommand{\textfraction}{0.1} \renewcommand{\floatpagefraction}{0.8} \newcommand{\Rm}{\operatorname{Rm}} \newcommand{\Ric}{\operatorname{Ric}} \newcommand{\Scal}{\operatorname{Scal}} \newcommand{\vol}{\operatorname{vol}} \newcommand{\area}{\operatorname{area}} \newcommand{\diam}{\operatorname{diam}} \newcommand{\dis}{\operatorname{dis}} \newcommand{\Int}{\operatorname{Int}} \newcommand{\Fr}{\operatorname{Fr}} \newcommand{\Cone}{\mathsf{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\CP}{\mathbb{CP}} \newcommand{\eps}{\varepsilon} \newcommand{\tB}{\widetilde{B}} ``` ## 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 $00$, 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 $$ \begin{equation}\label{eq:lrrp}\tag{1.1} d_{GH}\bigl(B_{\pi_x^*g}(\tilde x,s),B_{\R^n}(0,s)\bigr)<\delta s \qquad(00$ by Colding’s volume convergence theorem [Col97]; the converse fails. Rong’s conjecture under the stronger condition reads as follows. **Conjecture ([Ron26b, Conjecture 0.6, case (0.6.1)]).** Let $(M_i,g_i)$ be closed $n$-manifolds. Assume that $\Ric_{g_i}\ge-(n-1)g_i$ and that all their points are $(\delta,\rho)$-LRRP points. Suppose that $(M_i,g_i)$ converges in the Gromov–Hausdorff topology to a compact metric space $X$ of positive diameter and that $\vol_{g_i}(M_i)\to0$. Then, for $i$ large, there is a singular nilpotent fibration $f_i:M_i\to X$. Rong uses the unpointed distance in (1.1), which changes $\delta$ by at most a factor of two (Section 2.1). Rong’s definition requires $\delta\le\delta(n)$; we prove the conjecture for $\delta\le\delta_1(n)$ and show that the conclusion fails in general for $\delta>11/12$. Under the weaker LRVP hypothesis the picture is different. The earlier proposals under a Ricci lower bound and uniform LRVP all expect, at a minimum, that the convergence is realized by continuous maps $f_i:M_i\to X$ whose approximation errors tend to zero. Case (0.6.2) of Conjecture 0.6 in [Ron26a] asks in addition for finite infranil-quotient fibres capturing the short loops. Open Problem 4.4 of [Ron26b] fixes $\eps,\rho_0>0$, writes $X_\eps$ for the $\eps$-regular region and puts $$ \begin{equation*} X_0:=\{x\in X_\eps:d(x,X\setminus X_\eps)\ge\rho_0\}. \end{equation*} $$ A smooth infranil bundle is already available over $X_0$, and the problem asks whether it is the restriction of such a map. Our examples show that under LRVP the continuous approximation itself need not exist, so both statements fail regardless of their requirements on the fibres or over $X_0$. ### 1.1 Main Results Uniform LRRP supplies what ordinary short-time existence on a closed manifold lacks: control independent of the initial curvature. **Theorem 1.2 (Uniform Lifetime on Closed Manifolds).** Fix $n\ge2$, $\rho>0$ and $\alpha\in(0,1)$. There are $\delta_0=\delta_0(n,\alpha)>0$ and $T_0=T_0(n,\rho,\alpha)>0$ with the following property. If $(M^n,g_0)$ is closed, $\Ric_{g_0}\ge-(n-1)g_0$, and every point of $M$ is a $(\delta_0,\rho)$-LRRP, then the maximal Ricci flow $g(t)$ with $g(0)=g_0$ satisfies $$ \begin{equation}\label{eq:main-bound}\tag{1.2} T_{\max}>T_0,\qquad\sup_M|\Rm_{g(t)}|\le\frac{\alpha}{t}\quad(00$ 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(00$ 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)\to0$. Then there are $\eps_i\to0$ 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\downarrow0$ 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}\ge0$, 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)\to0$. - (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\le0.44$ 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. ### 1.2 Mechanism 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$, $\alpha0$, 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 $00$, 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*} $$ over $s'\in(0,s]$ and $\psi\in W^{1,2}_0(B)$ with $\psi\ge0$ and $\int_B\psi^2\,d\vol_h=1$. Wang’s pseudolocality theorem turns a lower bound for this entropy into a scale-invariant curvature bound; Section 4 uses it to give the comparison flow its Type-I bound. **Theorem 2.2 (Wang’s Pseudolocality Theorem [Wan20, Theorem 1.2]).** For every positive $\beta$ smaller than $1/(100n)$ there is $\delta_W=\delta_W(n,\beta)>0$ with the following property. Let $h(t)$, $0\le t\le T$, be a complete Ricci flow with bounded curvature on an $n$-manifold, and let $x$ be a point of the manifold. If $$ \begin{equation*} \nu\bigl(B_{h(0)}(x,\delta_W^{-1}\sqrt T),h(0),T\bigr)\ge-\delta_W^2, \end{equation*} $$ then $t\,|\Rm_{h(t)}|\le\beta$ on $B_{h(t)}(x,\beta^{-1}\sqrt t)$ for every $t\in(0,T]$. ### 2.4 Ricci–Harmonic Comparison Inputs We use the Ricci–harmonic gauge method of [Lee26, Sections 2–3]. The following proposition states the local versions needed here; Appendix A supplies the existence and derivative proofs. Derivatives of a solution $\partial_tF=\Delta_{g(t),h(t)}F$ use $g(t)$ and $h(t)$; when $F_t$ is a diffeomorphism onto its image, $\widehat g=(F_t^{-1})^*g$ is Ricci–DeTurck relative to $h$. **Proposition 2.3 (Local Comparison Estimates).** All source and target manifolds have dimension $n$. Let $\eps_{\mathrm{DT}}(n)\in(0,10^{-2}]$ be a threshold below which the discrepancy inequality (2.1) below holds, and set $\eps_{\mathrm L}(n):=\eps_{\mathrm{DT}}(n)/10$. - (i) Local Existence. For every $\Lambda_0>0$ there is $T_{\mathrm{ex}}=T_{\mathrm{ex}}(n,\Lambda_0)>0$ with the following property. Let $U\Subset M$ be a smooth domain, let a possibly incomplete Ricci flow $g(t)$ be smooth on a neighbourhood of $\overline U\times[0,T]$, let $N$ carry a complete Ricci flow $h(t)$, and suppose, on $[0,T]$, that for some $r>0$ $$ \begin{equation*} \sup_U|\Rm_g|+\sup_N|\Rm_h|\le r^{-2},\qquad\sup_U|\nabla\Rm_g|\le r^{-3}, \end{equation*} $$ while all curvature derivatives of $h(t)$ are globally bounded on $[0,T]$. If $f:U\to N$ is smooth on a neighbourhood of $\overline U$ and $|df|^2\le\Lambda_0$, then the Ricci–harmonic map heat flow starting from $f$ exists, with $|dF|^2\le10\Lambda_0$, on $$ \begin{equation*} U_r:=\{x\in U:B_{g(0)}(x,r)\Subset U\} \quad\hbox{for}\quad0\le t<\min\{T,T_{\mathrm{ex}}r^2\}. \end{equation*} $$ The proof is Lemma A.1. - (ii) Local Derivatives. Given $A,\Lambda_0>0$ and an integer $q\ge0$, there are $S_q=S_q(n,A,\Lambda_0,q)>0$ and $K_q=K_q(n,A,\Lambda_0,q)<\infty$ such that the following holds. If the Ricci flows $g(t),h(t)$ satisfy $B_{g(0)}(x,4r)\Subset M$, the target flow is complete, and $$ \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*} $$ for $00$ and an integer $L\ge1$ there is a constant $T_L^{\mathrm{dec}}>0$, depending only on $n$, $A$, $K$ and $L$, with the following property. Let $h(t)$ be a smooth Ricci flow on $N$ with $B_{h(0)}(y,4r)\Subset N$ and $|\Rm_h|\le A/t$ there, and let $\widehat g$ be a smooth Ricci–DeTurck flow relative to $h$ on a neighbourhood of the closed reference cylinder $\overline{B_{h(0)}(y,4r)}\times[0,T]$, with $\widehat g(0)=h(0)$, satisfying there $$ \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*} $$ Then, on $B_{h(0)}(y,r)$ and for $0\max\{2J,b+1\}$, therefore gives uniformly $$ \begin{equation*} \sup_{B_{h(0)}(y,3/2)}|H|_h\le t^J \end{equation*} $$ for any prescribed integer $J$, on a time depending only on $n,A,K,J$. For $x\in B_{h(0)}(y,1)$, the shrinking-balls lemma [ST21, Lemma 2.2], applied to $B_{h(0)}(x,3/8)\Subset B_{h(0)}(y,3/2)$, gives $B_{h(t)}(x,1/4)\Subset B_{h(0)}(y,3/2)$ for a common smaller time. Along a unit-speed $h(t)$-geodesic of length $s=t^{(J+1)/2}\le1/4$, which exists because $B_{h(t)}(x,1/4)$ is relatively compact, express $H$ in a parallel frame and apply Taylor’s formula to each component. This gives $$ \begin{equation}\label{eq:gradient-decay-interpolation}\tag{2.2} |\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} $$ Choose $J=L+2$ and decrease the time to absorb the spare factor of $t$ in the square of (2.2). Restoring $r$ gives the two bounds.∎ Section 5 applies these estimates on successive time slabs. ## 3 The Local Improvement and the First-Time Argument We state the local improvement for incomplete flows and derive Theorem 1.2; Sections 4 to 6 prove the local statement. **Proposition 3.1 (Local Type-I Improvement).** Fix $n\ge2$, $A\ge1$ and $0<\alpha0$, $\kappa_*=\kappa_*(n,A,\alpha)>0$, and $\Theta=\Theta(n,A,\alpha)\in(0,1)$ with the following property. Let $g(t)$, $0\le t0$, and assume that - (a) the closed ball $\overline{B_{g(0)}(p,4r)}$ is compact; - (b) on $B_{g(0)}(p,4r)\times(0,T)$, $$ \begin{equation}\label{eq:hyp-typeI}\tag{3.1} |\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}\tag{3.2} \Ric_{g(0)}\ge-(n-1)\kappa_*r^{-2}g(0);\tag{3.2} \end{equation} $$ - (d) $$ \begin{equation}\label{eq:hyp-gh}\tag{3.3} 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}\tag{3.4} |\Rm_{g(t)}|(p)\le\frac{\alpha}{t} \quad\bigl(00$. Before $T_A$, every lifted flow satisfies Proposition 3.1 with $p=\tilde x$ and $r=r_0$, hence $$ \begin{equation}\label{eq:ft-improved}\tag{3.7} t\sup_M|\Rm_{g(t)}|\le\alpha\Theta r_0^2,\qquad T_A>\Theta r_0^2. \end{equation*} $$ One final application at every lifted point gives (1.2) with $T_0:=\frac12\Theta r_0^2$.∎ ## 4 A 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. ### 4.1 Constants 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 $$ \begin{equation}\label{eq:deltaI}\tag{4.1} \frac{1-\delta_I}{(1+2\theta)^{n(n-1)}}\ge1-\frac12\delta_P.\tag{4.1} \end{equation} $$ All these constants depend only on $n$ and $\alpha$. ### 4.2 Initial 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 $$ \begin{equation}\label{eq:iso-hyp}\tag{4.2} \Ric_g\ge-(n-1)\kappa(\eta)r^{-2}g\ \hbox{ on }B_g(p,2r),\qquad d_{GH}\bigl(B_g(p,2r),B_{\R^n}(0,2r)\bigr)\le\delta(\eta)r .\tag{4.2} \end{equation} $$ Then, with $R:=c(\eta)r$, $$ \begin{equation}\label{eq:scal-R}\tag{4.3} \Scal_g\ge-R^{-2}\quad\hbox{on }B_g(p,R),\tag{4.3} \end{equation} $$ and every regular domain $E\Subset B_g(p,R)$ satisfies $$ \begin{equation}\label{eq:iso}\tag{4.4} \bigl(\area_g\partial E\bigr)^n \ge(1-\eta)n^n\omega_n \bigl(\vol_gE\bigr)^{n-1}.\tag{4.4} \end{equation} $$ *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 $$ \begin{equation*} C_{0,n}(r_V+\eta_V+\varepsilon_V) \le1-(1-\eta)^{1/n}. \end{equation*} $$ 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 $\Ric\ge-(n-1)\kappa_C$ on $B(y,r_C)$ 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 $\Psi(\delta\mid n)\to0$ 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 $$ \begin{equation}\label{eq:cI}\tag{4.5} c_I:=c(\delta_I),\qquad\delta_*:=\delta(\delta_I),\qquad\kappa_*:=\kappa(\delta_I),\tag{4.5} \end{equation} $$ the constants of Proposition 3.1, all depending only on $n,\alpha$. ### 4.3 Conformal 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 $C_\chi\ge1$ 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 $$ \begin{equation}\label{eq:cutoff}\tag{4.6} R|\nabla\chi|+R^2|\Delta\chi|\le C_\chi.\tag{4.6} \end{equation} $$ 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 $u\ge0$ on $\Omega$ 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 $$ \begin{equation}\label{eq:iso-P}\tag{4.7} \bigl(\area_{h_0}\partial E\bigr)^n \ge(1-\delta_P)n^n\omega_n\bigl(\vol_{h_0}E\bigr)^{n-1}.\tag{4.7} \end{equation} $$ *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 $\varphi=0$ on $B_g(p,R/3)$, $0\le\varphi<1$, $\varphi\to1$ at $\partial\Omega$, and (4.6) gives $$ \begin{equation}\label{eq:phi-bounds}\tag{4.8} R|\nabla\varphi|+R^2|\Delta\varphi|\le C_\varphi,\tag{4.8} \end{equation} $$ 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 $$ \begin{equation}\label{eq:w-deriv}\tag{4.9} e^{-w_\theta}w_\theta'\le C\theta^{-1},\qquad e^{-2w_\theta}\bigl(w_\theta''+(w_\theta')^2\bigr) \le C\theta^{-2}.\tag{4.9} \end{equation} $$ For $1-3\theta/20$, 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, $$ \begin{equation}\label{eq:iso-h0}\tag{4.11} \frac{\bigl(\area_{h_0}\partial E\bigr)^n}{\bigl(\vol_{h_0}E\bigr)^{n-1}}\ge(1-\delta_I)(1+2\theta)^{-n(n-1)}n^n\omega_n \ge(1-\tfrac12\delta_P)n^n\omega_n >(1-\delta_P)n^n\omega_n.\tag{4.11} \end{equation} $$ 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 $h_0$. ### 4.4 The 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 $$ \begin{equation}\label{eq:selected-bound}\tag{4.12} |\Rm_{h(t)}|\le\frac{\beta}{t} \quad(00). \end{equation*} $$ 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 $$ \begin{equation*} \nu\bigl(B_{h_0}(x,\delta_W^{-1}\sqrt T),h_0,T\bigr) \ge-\tfrac12\delta_W^2-\eps_P^2=-\tfrac34\delta_W^2>-\delta_W^2 . \end{equation*} $$ Thus Theorem 2.2 applies at every point and gives $$ \begin{equation}\label{eq:selected-T}\tag{4.13} |\Rm_{h(t)}|\le\frac{\beta}{t} \quad(00$, and put $D_j:=B_{g(0)}(p,j\ell)$. Suppose $D_{64}\Subset N$, and let $h(t)$, $0\le t\le T$, be a complete smooth Ricci flow on $N$. Assume $$ \begin{equation}\label{eq:initial-order}\tag{5.1} 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}\tag{5.2} \sup_{D_{64}}|\Rm_{g(t)}|+\sup_N|\Rm_{h(t)}| \le\frac{A_*}{t}.\tag{5.2} \end{equation} $$ For every integer $L\ge1$ there are constants $\tau_L>0$, $C_{\mathrm{disp}}$ and $C_L<\infty$, depending only on $n$, $A_*$ and $L$, with the following property. There is a Ricci–harmonic map heat flow $F$ from $D_8\times[0,\min\{T,\tau_L\ell^2\})$ to $N$ with $F_0=\mathrm{id}$ such that, for $02a_*+2$; it is the decay exponent in (5.20) below, and the bound on $m$ makes the first integral in (5.21) converge. A time slab beginning at a time $t_k>0$ is handled with the two scales $$ \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}\tag{5.8} \mu0$ and $\mathfrak r_k>0$ put $$ \begin{equation*} B_{k,j}:=B_{g(0)}\bigl(p,\mathfrak r_k-j\Lambda\sqrt{t_k}\bigr) \quad(j\ge0). \end{equation*} $$ 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) $$ \begin{equation}\label{eq:ind-equiv}\tag{5.10} (1-\eps_*)g(t)\le F_t^*h(t)\le(1+\eps_*)g(t);\tag{5.10} \end{equation} $$ - (III) for $x\in B_{k,5}$ and $0\le t\le t_k$, $$ \begin{equation}\label{eq:ind-disp}\tag{5.11} \begin{split} d_{g(0)}(x,F_t(x))&\le\frac14\Lambda\sqrt{t_k}, \\ B_{g(0)}(x,\Lambda\sqrt{t_k}) &\subset F_t\bigl(B_{g(0)}(x,2\Lambda\sqrt{t_k})\bigr). \end{split}\tag{5.11} \end{equation} $$ 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](https://kspectra.ai/papers/collapsing-manifolds-local-rewinding-reifenberg/figs/fig_slabs.svg) *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.* ### 5.2 The 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 $00$ and $0\le t\le t_{k+1}$, $$ \begin{equation}\label{eq:shrink}\tag{5.12} B_{g(t)}(x,c\sqrt{t_k}) \subset B_{g(0)} \!\left(x,(c+C_n\sqrt{A_*})\sqrt{t_k}\right)\tag{5.12} \end{equation} $$ 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, $$ \begin{equation}\label{eq:shi}\tag{5.13} |\nabla^j\Rm_g|+|\nabla^j\Rm_h| \le C_jt_k^{-1-j/2}.\tag{5.13} \end{equation} $$ Choose a smooth domain $B_k'$ with $$ \begin{equation*} B_{g(0)}\bigl(p,\mathfrak r_k-\tfrac12\Lambda\sqrt{t_k}\bigr) \Subset B_k'\Subset B_{k,0}. \end{equation*} $$ 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 $\Lambda\ge2(c_*+C_n\sqrt{A_*})$. 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 $t_{k+1}-t_k\le\mu t_kC(n,A_*)$ 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 $\eps'\downarrow0$ handles non-strict initial inequalities. Apply (5.16) with $\eps=\eps_*$. Since $C_{\mathrm{bar}}(s-t_k)/t_k\le C_{\mathrm{bar}}\mu\le\frac23\eps_*$ by (5.8), this gives $$ \begin{equation}\label{eq:rough}\tag{5.18} (1-3\eps_*)g(s)\le\check h(s)\le(1+3\eps_*)g(s)\tag{5.18} \end{equation} $$ on $B_{k,2}\times[t_k,t_{k+1}]$, 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 $$ \begin{equation}\label{eq:traj}\tag{5.19} \dot x_i(s)=-(dF_s|_{x_i(s)})^{-1} (\partial_sF_s)(x_i(s)),\qquad x_i(s_0)=x_i.\tag{5.19} \end{equation} $$ 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 $4\sigma_k'=\Lambda\sqrt{t_k}/2$, $$ \begin{equation*} \overline{B_{h(0)}(y,4\sigma_k')} \subset B_{g(0)}\left(x,\tfrac34\Lambda\sqrt{t_k}\right) \Subset B_{g(0)}(x,\Lambda\sqrt{t_k}). \end{equation*} $$ 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 $B_{g(0)}(z,4\sigma_k')\subset B_{g(0)}(x,\tfrac52\Lambda\sqrt{t_k})\subset B_{k,2}$. So Proposition 2.3(iii), applied at every such $z$ with the radius $\sigma_k'$ and $q=2$, gives $|(\nabla^h)^2\widehat g_{s'}|_h\le K_2^{\mathrm{DT}}/s'$ on $\overline{B_{h(0)}(y,4\sigma_k')}\times(0,t_k]$, 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 $00$ 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 $\delta_{\mathrm{met}}$ at $s=t_k$. 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 $C_{\mathrm{bar}}\mu\le\frac23\eps_*$, 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 $$ \begin{equation*} |\partial_sF_s|_{h(s)} \le C t_k^{-(m+1)/2}s^{m/2},\qquad0R_k \quad(z\in\partial\mathcal B), \end{equation*} $$ 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 $(\mathrm{Ind}_{k+1})$.∎ ### 5.3 Proof 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 $k\ge1$ with $t_{k-1}<\min\{T,\tau_L\}$ and $t_k24$. 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 $t$: $(31,t_0)$ if $t\le t_0$, and, if $t_{k-1}t_0$ 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 $$ \begin{equation*} |(\nabla^h)^j\widehat g|_h\le K_4^{\mathrm{DT}}t^{-j/2} \qquad(1\le j\le4)\quad\hbox{on }D_8, \end{equation*} $$ 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 $n,A_*,L$.∎ ## 6 Completion 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 $h(t)$ of $h_0$ on $[0,T_H]$, 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 $u\ge0$ on $N$, the conformal factor is at least one and $h(0)\ge g(0)$ as quadratic forms on all of $N$; and $u=0$ on $B_{g(0)}(p,R/4)$, so $h(0)=g(0)$ on $D_{64}$. This is (5.1). Moreover $D_{64}\Subset N\Subset B_{g(0)}(p,3R/4)\Subset U$. By (3.1) and (4.12), $$ \begin{equation}\label{eq:boot-curv}\tag{6.1} \sup_{D_{64}}|\Rm_{g(t)}|+\sup_N|\Rm_{h(t)}|\le\frac{A+\beta}{t} \qquad\bigl(00$ so that the right-hand side of (6.2) is at most $\alpha$ whenever $\tau\le\tau_0$, and also $$ \begin{equation}\label{eq:tau0}\tag{6.3} \tau_0\le\min\left\{1,\tau_4,512^2(\eps_Pc_H)^2, \tfrac14C_{\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*} $$ which proves (3.5).∎ ## 7 Complete Noncompact Initial Manifolds We now remove compactness of the initial manifold and prove Theorem 1.3. Throughout this section $(M^n,g_0)$ is complete and noncompact, $\Ric_{g_0}\ge-(n-1)g_0$, and every point of $M$ is a $(\delta_{\mathrm{nc}},\rho)$-LRRP, where $\delta_{\mathrm{nc}}$ is fixed in (7.1). We choose conformal completions whose initial metrics agree with $g_0$ on an exhaustion of $M$. Uniform estimates for their Ricci flows on one common time interval give a locally smooth diagonal limit, and the agreement of the initial metrics with $g_0$ on the exhaustion identifies the initial value of that limit with $g_0$. ### 7.1 Constants The conformal change costs a factor close to one in the isoperimetric constant, so the collar estimate needs slightly more room than was used in Section 4, and the constants are fixed accordingly. Let $\beta$, $\eps_P$, $\theta$ and $\delta_I$ be as in Section 4.1, and let $w_\theta$ be the conformal profile of Lemma 4.2. Put $\eta_0:=\delta_I/2$, and let $$ \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}\tag{7.1} \delta_{\mathrm{nc}}:=\min\{\delta_0,\delta_\flat/2\}.\tag{7.1} \end{equation} $$ All these constants depend only on $n$ and $\alpha$. ### 7.2 The Conformal Exhaustion The first lemma constructs the exhaustion and, for every $j$ and every point of $\Omega_j$, a comparison flow on the local cover. The conformal change pushes the artificial boundary to infinity, while isoperimetric control on the local cover survives at the scale $e^{-u_j(x)}r_*$ of the original metric, which becomes a fixed scale for the conformal metric. This scale tends to zero towards $\partial\Omega_j$, so the construction uses the LRRP condition at all scales below $\rho$. **Lemma 7.1 (Uniform Conformal Exhaustion).** There are relatively compact domains $\Omega_j\Subset M$ with smooth boundary, compact sets $E_j\subset\Omega_j$ whose union is $M$, and smooth functions $u_j\ge0$ on $\Omega_j$ such that $h_{j,0}:=e^{2u_j}g_0$ is a complete metric of bounded curvature on $\Omega_j$ and $$ \begin{equation}\label{eq:exhaustion}\tag{7.2} u_j=0\ \hbox{on }E_j,\qquad\overline{\Omega_j}\subset\Int E_{j+1}.\tag{7.2} \end{equation} $$ Moreover, there are $r_E,T_E>0$, depending only on $n,\rho,\alpha$, with the following property. For every $j$ and every $x\in\Omega_j$, let $\widetilde h_{j,0}:=\pi_x^*h_{j,0}$ on the part of the local cover $\tB_{g_0}(x,\rho)$ that lies over $\Omega_j$. There are an open set $N_{j,x}$ in this part and a complete bounded-curvature Ricci flow $h^\sharp_{j,x}(t)$ on $N_{j,x}\times[0,T_E]$ such that, with $\ell_E:=r_E/512$, $$ \begin{equation}\label{eq:selected-E}\tag{7.3} \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(0a_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 $$ \begin{equation}\label{eq:phi-j}\tag{7.4} |\nabla\varphi_j|+|\Delta\varphi_j|\le C(n).\tag{7.4} \end{equation} $$ 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 $$ \begin{equation*} \overline{\Omega_j}\subset\overline{B_{g_0}(o,2j+\tfrac12)}\subset\Int E_{j+1}, \end{equation*} $$ 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 $$ \begin{equation}\label{eq:CE}\tag{7.5} |\nabla^{h_{j,0}}u_j|_{h_{j,0}}\le C_E,\qquad\Scal_{h_{j,0}}\ge-C_E.\tag{7.5} \end{equation} $$ 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), $$ \begin{equation*} |\nabla^{g_0}e^{-u_j}|_{g_0}=|\nabla^{h_{j,0}}u_j|_{h_{j,0}} \le C_E. \end{equation*} $$ So $e^{-u_j}$ is $C_E$-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 $cC_Er_*\le\frac16$, $$ \begin{equation}\label{eq:osc}\tag{7.6} \overline{B_{g_0}(x,cs)}\Subset\Omega_j,\qquad\bigl|e^{u_j(x)-u_j}-1\bigr|\le cC_Er_*\ \hbox{ on }B_{g_0}(x,cs).\tag{7.6} \end{equation} $$ *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 $\pi_x^*g_0$ at $\tilde x$, with $\eta=\eta_0$ and $r=s$: every regular domain $E\Subset B_{\pi_x^*g_0}(\tilde x,c_0s)$ satisfies $$ \begin{equation}\label{eq:iso-lift}\tag{7.7} \bigl(\area_{\pi_x^*g_0}\partial E\bigr)^n \ge(1-\eta_0)n^n\omega_n \bigl(\vol_{\pi_x^*g_0}E\bigr)^{n-1}.\tag{7.7} \end{equation} $$ *The Conformal Metric on the Cover.* Put $\eps_E:=c_0C_Er_*$ and require $\eps_E\le\frac16$. 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 $E\Subset B_{\pi_x^*g_0}(\tilde x,c_0s)$, $$ \begin{equation}\label{eq:iso-conf}\tag{7.8} \bigl(\area_{\widetilde h_{j,0}}\partial E\bigr)^n \ge\Bigl(\frac{1-\eps_E}{1+\eps_E}\Bigr)^{n(n-1)}(1-\eta_0)n^n\omega_n \bigl(\vol_{\widetilde h_{j,0}}E\bigr)^{n-1}.\tag{7.8} \end{equation} $$ 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 $C_Er_E^2\le1$. 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) $$ \begin{equation}\label{eq:scal-E}\tag{7.9} \Scal_{\widetilde h_{j,0}}\ge-r_E^{-2} \quad\hbox{on }B_{\widetilde h_{j,0}}(\tilde x,r_E),\tag{7.9} \end{equation} $$ 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 $$ \begin{equation}\label{eq:cut-bounds}\tag{7.10} r_E|\nabla^{\widetilde h_{j,0}}\tilde\chi|_{\widetilde h_{j,0}} +r_E^2|\Delta_{\widetilde h_{j,0}}\tilde\chi|\le C_\chi',\tag{7.10} \end{equation} $$ with $C_\chi'=C_\chi'(n,\alpha)$. By the two-sided bound for $e^{u_j}$ on $B_{\pi_x^*g_0}(\tilde x,c_0s)$, 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 $\eps_E\le\frac16$. Hence $$ \begin{equation}\label{eq:cut-incl}\tag{7.11} \tilde\chi=1\ \hbox{on }B_{\widetilde h_{j,0}}(\tilde x,r_E/3),\qquad\operatorname{supp}\tilde\chi\subset B_{\widetilde h_{j,0}}(\tilde x,3r_E/4).\tag{7.11} \end{equation} $$ *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.∎ ### 7.3 Uniform 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 $$ \begin{equation}\label{eq:lifetime}\tag{7.12} \sup_{\Omega_j}|\Rm_{h_j(t)}|\le\frac{\alpha}{t} \quad(00$ so that the right-hand side is at most $\alpha$ for $\tau\le\tau_E$ and $\tau_E\le\min\{1,\tau_4,\frac14C_{\mathrm{disp}}^{-2},(2C_{\mathrm{met}})^{-1}\}$, and put $$ \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}\tag{7.13} t|\Rm_{h_j(t)}|(x)\le\alpha\quad\bigl(0T_E'$, and (7.12) follows.∎ ### 7.4 The 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 $$ \begin{equation*} \sup_M|\Rm_{g(t)}|\le\frac{\alpha}{t} \qquad(00$ and $\dim\bar X0$ and $\vartheta\in(0,1)$, and put $$ \begin{equation*} \alpha:=\frac{\vartheta}{2\sqrt n\,(n-1)},\qquad\delta_1:=\delta_0(n,\alpha), \end{equation*} $$ where $\delta_0$ is the constant of Theorem 1.2. There is $\bar t=\bar t(n,\rho,\vartheta)\in(0,1]$ with the following property. Let $(M_i,g_i)$ be closed $n$-manifolds with $\Ric_{g_i}\ge-(n-1)g_i$ and uniform $(\delta_1,\rho)$-LRRP, converging in the Gromov–Hausdorff topology to a compact metric space $X$ of positive diameter, with $\vol_{g_i}(M_i)\to0$. Let $g_i(t)$ be the Ricci flow with $g_i(0)=g_i$, and put $\bar g_i:=(\alpha/\bar t)\,g_i(\bar t)$. Then every subsequence has a further subsequence along which the following hold. - (a) The sequence $(M_i,\bar g_i)$ satisfies the hypotheses of Theorem 8.1 for some limit $\bar X$. Let $Y$, $\hat f_i$, and $\bar f_i:M_i\to\bar X=Y/O(n)$ be as in that theorem. - (b) There is a homeomorphism $\Phi:X\to\bar X$, the limit of the identity maps $(M_i,d_{g_i})\to(M_i,d_{\bar g_i})$, such that $\Phi$ and $\Phi^{-1}$ are Hölder continuous, with exponents $1-\vartheta$ and $1/(1+\vartheta)$ and with constants depending only on $n,\rho,\vartheta$, and $\diam X$. - (c) The maps $$ \begin{equation*} f_i:=\Phi^{-1}\circ\bar f_i:M_i\longrightarrow X \end{equation*} $$ are continuous and surjective, every fibre of $f_i$ is an infranilmanifold of positive dimension, and $f_i$ is an $\eps_i'$-Gromov–Hausdorff approximation from $(M_i,d_{g_i})$ to $X$, with $\eps_i'\to0$. Consequently, for every $\eps>0$, all but finitely many $M_i$ of the original sequence carry a map $f_i=\Phi^{-1}\circ\bar f_i$ of this kind that is an $\eps$-Gromov–Hausdorff approximation from $(M_i,d_{g_i})$ to $X$, with $\hat f_i$ satisfying (i)–(iii) of Theorem 8.1 for $\eps$ in place of $\eps_i$. Here $Y$, $\bar X$, and $\Phi$ may depend on $i$, and the constants in (b) do not. Thus $f_i$ fits into (8.1), with $X$ identified with $Y/O(n)$ through $\Phi$. The estimates of Theorem 8.1 use the canonical metric of $\bar g_i$; the $g_i$- and $\bar g_i$-frame bundles are $O(n)$-equivariantly diffeomorphic via $b\mapsto b\,(b^*\bar g_i)^{-1/2}$. ### 8.3 Uniform Bi-Hölder Distortion The Ricci flow enters only through this comparison at one fixed time. **Lemma 8.3 (Uniform Bi-Hölder Distortion).** Let $n,\rho,\vartheta,\alpha,\delta_1$ be as in Theorem 8.2. There are $\bar t,\bar r\in(0,1]$ and $\bar C\ge1$, depending only on $n,\rho,\vartheta$, with the following property. Let $(M^n,g_0)$ be closed with $\Ric_{g_0}\ge-(n-1)g_0$ and uniform $(\delta_1,\rho)$-LRRP, let $g(t)$ be its Ricci flow, and put $\bar g:=(\alpha/\bar t)\,g(\bar t)$, $d:=d_{g_0}$, and $\bar d:=d_{\bar g}$. Then $|\sec_{\bar g}|\le1$, $\vol_{\bar g}(M)\le\bar C\vol_{g_0}(M)$, and for all $x,y\in M$ $$ \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}\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}\tag{8.3} \\ \bar d(x,y)&\le\bar C\bigl(d(x,y)+1\bigr). \label{eq:dist-coarse}\tag{8.4}\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(00$. Thus Theorem 8.1 applies and gives (a). In particular the maps $\bar f_i$ are continuous, surjective $\eps_i$-Gromov–Hausdorff approximations from $(M_i,\bar d_i)$ to $\bar X$, 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 $$ \begin{equation}\label{eq:lift-points}\tag{8.6} \bigl|d_i(x_i(z),x_i(z'))-d_X(z,z')\bigr|<3\eps_i \qquad(z,z'\in X).\tag{8.6} \end{equation} $$ 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 $$ \begin{align} d_{\bar X}(\Phi(z),\Phi(z'))&\le\bar C\,d_X(z,z')^{1-\vartheta},\label{eq:Phi-upper}\tag{8.7}\tag{8.7} \\ d_X(z,z')&\le\bar C\,d_{\bar X}(\Phi(z),\Phi(z'))^{1/(1+\vartheta)}.\label{eq:Phi-lower}\tag{8.8}\tag{8.8}\end{align} $$ 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 $$ \begin{equation}\label{eq:Phi-close}\tag{8.9} \sup_{x\in M_i}d_{\bar X}\bigl(\Phi(\psi_i(x)),\bar f_i(x)\bigr) \longrightarrow0.\tag{8.9} \end{equation} $$ 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 $\eps$ on $X'_\eps$. 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 $\eps\downarrow0$ 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 $$ \begin{equation*} \sup_{x\in M_i}d_X\bigl(\psi_i(x),f_i(x)\bigr)\longrightarrow0. \end{equation*} $$ Hence $f_i$ is an $\eps_i'$-Gromov–Hausdorff approximation, where $\eps_i'$ is $\eps_i$ 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 $\vartheta=\frac12$, so that $\delta_1(n)=\delta_0\bigl(n,(4\sqrt n\,(n-1))^{-1}\bigr)$. By its last assertion there are indices $i_10$ 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 $0<{\mathfrak q}<2\cdot10^{-6}$. 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 $W_i$, $i\ge1$, 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 $y\in\Sigma_\infty$, $$ \begin{equation}\label{eq:wedge}\tag{9.1} \varpi_i(y)=*\quad\hbox{or}\quad\varpi_k(y)=*.\tag{9.1} \end{equation} $$ Indeed, for $y_j\to y$, one of the two disjoint cores is avoided along a subsequence, and the corresponding quotient converges to $*$. Also $$ \begin{equation}\label{eq:Wi}\tag{9.2} \mathcal W_i:=\varpi_i^{-1}(Q_i\setminus\{*\})=\iota_i(\Int W_i).\tag{9.2} \end{equation} $$ 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 $s_m\downarrow0$ and a constant $C_\Sigma\ge1$ 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 $\frac43s_i$ 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 $\frac43s_i$ 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 $\mathsf f_j\le2s_m$ 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 $\mathsf f_j=\frac43s_i\ge\frac43s_m$, 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 $\mathsf f_j\le32\,d_{\Sigma_j}(p,\cdot)$, 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 $C_\Sigma=2C_*+1$.∎ ### 9.2 The Products Let $\lambda_j\downarrow0$, 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) $\Ric_{P_j}\ge0$. - (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 $\vol(P_j)\to0$. *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 $\Ric_{g_j}\ge0$ gives $$ \begin{equation}\label{eq:allpoint}\tag{9.3} \vol B_{\Sigma_j}(z,\tfrac13)\ge\frac{\vol B_{\Sigma_j}(z,\mathfrak d+1)}{3^4(\mathfrak d+1)^4} \ge\frac{v_0}{3^4(\mathfrak d+1)^4}.\tag{9.3} \end{equation} $$ For $x=[z,0]$, let $B'$ be the component over $B_{P_j}(x,1)$ in $\Sigma_j\times\R$ containing $(z,0)$. Since $\lambda_j<2/3$, the connected cylinder $B_{\Sigma_j}(z,\frac13)\times\R$ lies in $B'$, and its rectangle with $|t|<\frac13$ 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 $$ \begin{equation*} \vol B_{\pi_x^*g}(\tilde x,1)\ge\vol\bigl(B_{\Sigma_j}(z,\tfrac13)\times(-\tfrac13,\tfrac13)\bigr) =\frac23\vol B_{\Sigma_j}(z,\tfrac13)\ge v_* . \end{equation*} $$ (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, $\vol(\Sigma_j)\le\omega_4\mathfrak d^4$.∎ ### 9.3 No 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 $0\le t\le1$ 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 $b(0)$: $$ x\longmapsto\frac{b(x)-b(-tx)}{|b(x)-b(-tx)|}. $$ At $t=1$ 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 $00$ 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 $$ \begin{equation*} \sqrt{c_1}\,|x-y|\le d_{\Sigma_j}(\phi_j^{(1)}(x),\phi_j^{(1)}(y))\le\sqrt{c_2}\,|x-y|, \qquad B_{\Sigma_j}\bigl(\phi_j^{(1)}(0),\tfrac12\sqrt{c_1}\,r_2\bigr) \subset\phi_j^{(1)}\bigl(B_{\R^4}(0,\tfrac12r_2)\bigr), \end{equation*} $$ 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 $$ \begin{equation*} 00$, and let $P=M\times(\R/\lambda\Z)$ with the product metric. For every $x\in P$, every $\rho>0$ and every $0\frac{11}{12}$ and every $\rho>0$. *Proof.* We verify the hypotheses of Lemma 9.7 with $n=5$, $\eps=\frac16$, $m_1=10$, and $Z$ the ball of radius $s$ about $z_0=\tilde x$ in the local cover at $x$. For (a), the ten sets $\{y\in S^4:\pm y_i=\max_k|y_k|\}$ cover $S^4$ and have diameter at most $\sqrt{16/5}<11/6$: in the first set, $\langle y,y'\rangle\ge2/5-1$. For (b), the circle projection lifts to a $1$-Lipschitz function $\varsigma$ on $\tB_P(x,\rho)$ with $\varsigma(\gamma(t))=t$ along the arclength lift $\gamma$ from $\tilde x$, for $|t|<\rho$. Thus the lifted-circle points $\gamma(ks/6)$, $-5\le k\le4$, are pairwise $s/6$-separated and lie within $5s/6$ of $\tilde x$. For (c), Lemma 9.8 with $a=0.29$ gives $m_2\le12\,646$ sets. Finally, the $16\,875$ points $(s/6)k$ with $k\in\Z^5$ and $|k|\le5$ verify (d), since they are $s/6$-separated and lie in the ball of radius $5s/6$. Thus Lemma 9.7 gives $(1-1/12)s$.∎ ### 9.6 The Reifenberg Constant: Lower Bound A quantitative Borsuk–Ulam theorem separates the tangent model $\R^3\times\Cone(S^1_{2\pi{\mathfrak q}})$ from $\R^5$. **Lemma 9.10 (Local Covers of Products).** Let $\Sigma$ be a closed Riemannian manifold, $\lambda>0$, and $x=[z,0]\in\Sigma\times(\R/\lambda\Z)$. Let $00.4413\ \ \hbox{for }{\mathfrak q}\le\tfrac1{100},\qquad\delta_{\mathrm{low}}(\tfrac14)>0.37,\qquad\delta_{\mathrm{low}}(\tfrac12)>0.25 . \end{equation*} $$ *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 $|y|^2+\varrho^2\le1$, and the distance $d$ of two of them is given by $$ \begin{equation*} d^2=|y-y'|^2+\varrho^2+\varrho'^2-2\varrho\varrho'\cos\angle(c,c'), \qquad0\le\angle(c,c')\le\pi{\mathfrak q}\le\tfrac\pi2 , \end{equation*} $$ 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, $$ \begin{equation}\label{eq:Pi}\tag{9.4} d^2-2c_{\mathfrak q}\varrho(x)\varrho(x')\le|\Pi(x)-\Pi(x')|^2\le d^2 .\tag{9.4} \end{equation} $$ *Antipodes.* For $x$ and $-x$ the distance of the chosen points is at least $2-\eps$. Since the cosine above is nonnegative and $|y-y'|^2\le2|y|^2+2|y'|^2\le4-2\varrho^2-2\varrho'^2$, we get $(2-\eps)^2\le4-\varrho^2-\varrho'^2$, that is, $\varrho(x)^2+\varrho(-x)^2\le4\eps-\eps^2$. 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 $R^2\le0$, the inequality $4\eps^2\ge\frac52R^2$ 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 $S^4$ to $S^3$ 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 $\arccos(-\frac14)$: 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, $4\eps^2\ge\frac52R^2$. 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 $\delta\le0.44$ 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 $00$, and hence analogues of parts (i)–(iv) of Theorem 1.5 and of Corollary 1.6 in every dimension $n\ge5$; the obstruction follows by restriction to a slice. ## 10 Discussion ### 10.1 Between 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. ### 10.2 Neighbouring 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. ## A Local 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.1 Interior 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 $02s$, let $\mathsf k_s$ be a normalized smooth radial kernel on $|v|\ell_j/10$ 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 $$ \begin{equation}\label{eq:map-boundary-radius}\tag{A.2} \hat d\asymp\min\{d,1\},\qquad|\nabla\hat d|+\hat d|\nabla^2\hat d|\le C_n,\qquad|\nabla^q\hat d|\le C_q\hat d^{1-q}.\tag{A.2} \end{equation} $$ Also $\hat d=1/8$ on $U_1$. The constants $C_q$ for $q\ge3$ 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 $z\le1/2$ and $\upsilon_3(z)=0$ for $z\ge1$, and set $$ \begin{equation*} u=\upsilon_3(64\hat d),\qquad G(t)=e^{2u}g(t). \end{equation*} $$ Then $G\ge g$, $G=g$ on $U_1$, and $G(t)$ is complete because $e^u\asymp d^{-1}$ near $\partial U$. The conformal curvature formula gives $$ \begin{equation}\label{eq:map-completed-zero-order}\tag{A.3} |\Rm_G|_G+|\partial_tG|_G\le C_n \qquad(0\le t\le\min\{T,s_0\}).\tag{A.3} \end{equation} $$ Indeed $e^{-u}|du|+e^{-2u}|\nabla^2u|\le C_n$ and $|\partial_tG|_G=|\partial_tg|_g$. All the following bounds are finite: $$ \begin{equation}\label{eq:map-completed-higher}\tag{A.4} \sup_{U\times[0,\min\{T,s_0\}]} \bigl(|(\nabla^G)^j\Rm_G|_G+|(\nabla^G)^j\partial_tG|_G\bigr)<\infty\qquad(j\ge0).\tag{A.4} \end{equation} $$ 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 $|\Rm_h|\le1$, $(\partial_t-\Delta_G)e_0\le C_n(e_0+e_0^2)$. 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 $$ \begin{equation*} T_{\mathrm{ex}}=\min\left\{s_0,\frac1{2C_n} \log\frac{2(1+\Lambda_0)}{1+2\Lambda_0}\right\}>0. \end{equation*} $$ Let $b=\min\{T,T_{\mathrm{ex}}\}$. The bounds (A.4) and those of the target are uniform on $[0,b]$, and $|dF_{t'}|_{G,h}^2\le2\Lambda_0$ for $t'