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On noncollapsed $\mathbb{F}$-limit metric solitons

Pak-Yeung Chan, Zilu Ma, Yongjia Zhang

Abstract

A noncollapsed $\mathbb{F}$-limit metric soliton is a self-similar singularity model that inevitably arises when studying the Ricci flow with the tool of $\mathbb{F}$-convergence [Bam20a,Bam20b,Bam20c]. In this article, we shall present a systematic study of the noncollapsed $\mathbb{F}$-limit metric soliton, and show that, apart from the known results in [Bam20c], it satisfies many properties of smooth Ricci shrinkers. In particular, we show a quadratic lower bound for the scalar curvature, a local gap theorem, a global Sobolev inequality, and an optimal volume growth lower bound.

On noncollapsed $\mathbb{F}$-limit metric solitons

Abstract

A noncollapsed -limit metric soliton is a self-similar singularity model that inevitably arises when studying the Ricci flow with the tool of -convergence [Bam20a,Bam20b,Bam20c]. In this article, we shall present a systematic study of the noncollapsed -limit metric soliton, and show that, apart from the known results in [Bam20c], it satisfies many properties of smooth Ricci shrinkers. In particular, we show a quadratic lower bound for the scalar curvature, a local gap theorem, a global Sobolev inequality, and an optimal volume growth lower bound.
Paper Structure (26 sections, 40 theorems, 248 equations)

This paper contains 26 sections, 40 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Basic properties of the potential function and the volume
  3. A lower bound for the scalar curvature
  4. A local gap theorem
  5. Global nu-functional and Sobolev inequality
  6. Volume growth lower estimate
  7. Metric soliton as an F-limit
  8. Metric soliton as an ancient F-limit
  9. Center and the potential function
  10. Center of metric soliton
  11. Estimates of the potential function
  12. Scalar curvature near a center
  13. A lower bound for the scalar curvature
  14. Local lower bound of the scalar curvature near Hn-centers
  15. Proof of the scalar curvature lower bound
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(X,d,\nu)$ be the model of an $n$-dimensional noncollapsed $\mathbb{F}$-limit metric soliton and $(\mathcal{R}_X,g,f_0)$ the regular part of the model. Let $x_0\in\mathcal{R}_X$ be a center of the soliton. Then, for any $\varepsilon>0$, we have where $C(\varepsilon)$ is a constant depending only on $\varepsilon$.

Theorems & Definitions (66)

  • Theorem 1.1: Potential function estimate
  • Corollary 1.2: Volume ratio upper bound at a center
  • Corollary 1.3: Volume ratio upper bound
  • Theorem 1.4: Scalar curvature lower bound
  • Theorem 1.5: Local gap theorem
  • Proposition 1.6
  • Theorem 1.7
  • Corollary 1.8: Logarithmic Sobolev inequalities
  • Corollary 1.9: Sobolev inequality
  • Theorem 1.10
  • ...and 56 more