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An $ε$-regularity theorem for Perelman's reduced volume

Liang Cheng, Yongjia Zhang

TL;DR

The paper proves an ε-regularity theorem for Perelman’s reduced volume under Ricci flow: if the reduced volume $\\mathcal{V}_{x,t}(\tau)$ is close to its model value 1, then the curvature radius $r_{\mathrm{Rm}}(x,t)$ cannot be arbitrarily small. The authors overcome the lack of base-point control by normalizing counterexamples and applying Bamler’s $\mathbb{F}$-compactness to extract a limit flow, which they show is a Gaussian shrinker via vanishing $\mathcal{N}(\tau)$. Local uniform estimates for the reduced distance and its gradient are established, and together with the soliton structure of the limit, yield a contradiction that proves the ε-regularity result. The work also yields corollaries connecting near-maximal reduced volume to volume lower bounds, strengthening the link between monotonicity formulas and regularity in Ricci flow. Overall, the paper provides a robust regularity criterion for the reduced volume and develops new tools for handling nonlocal base-point dependence via Bamler’s limit theory.

Abstract

In this article, we prove an $ε$-regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to $1$, then the curvature radius at the base point cannot be too small.

An $ε$-regularity theorem for Perelman's reduced volume

TL;DR

The paper proves an ε-regularity theorem for Perelman’s reduced volume under Ricci flow: if the reduced volume is close to its model value 1, then the curvature radius cannot be arbitrarily small. The authors overcome the lack of base-point control by normalizing counterexamples and applying Bamler’s -compactness to extract a limit flow, which they show is a Gaussian shrinker via vanishing . Local uniform estimates for the reduced distance and its gradient are established, and together with the soliton structure of the limit, yield a contradiction that proves the ε-regularity result. The work also yields corollaries connecting near-maximal reduced volume to volume lower bounds, strengthening the link between monotonicity formulas and regularity in Ricci flow. Overall, the paper provides a robust regularity criterion for the reduced volume and develops new tools for handling nonlocal base-point dependence via Bamler’s limit theory.

Abstract

In this article, we prove an -regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to , then the curvature radius at the base point cannot be too small.

Paper Structure

This paper contains 11 sections, 18 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Perelman's $\mathcal{L}$-geometry
  5. Bamler's theory of non-collpased limits of Ricci flows
  6. local estimates for the reduced distance
  7. The proof of Theorem \ref{['main_thm']}
  8. Setting up the contradictory sequence
  9. Local uniform $C^{0,1}$-estimates
  10. Soliton structure on the limit
  11. Completion of the proof

Key Result

Theorem 1.1

There is a dimensional constant $\epsilon(n)>0$ with the following property. Let $(M^n, g_t)_{t\in I}$ be a complete Ricci flow with bounded curvature within each compact time interval. Let $(x,t)\in M\times I$ be a space-time point and $r>0$ a scale with $[t-2r^2,t]\subset I$. If, furthermore then we have $r_{\operatorname{Rm}} (x, t) \ge r$; here $\mathcal{V}$ and $r_{\mathop{\mathrm{Rm}}\noli

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Definition 2.1
  • Theorem 2.2: Proposition 2.7, Proposition 2.11, and Lemma 2.14 in Y1
  • Theorem 2.3: P1, see also Lemma 2.19 and Theorem 2.20 in Y1
  • Lemma 2.4: Perelman P1
  • Theorem 2.5: PerelmanP1, see also Theorem 4.3 and Theorem 4.5 in Y1
  • Theorem 2.6: Bam20c, Bam20b,Bam20b
  • Theorem 2.7: Bam20b, Bam20c
  • ...and 23 more