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Ancient Ricci flows with nonnegative Ricci curvature

Yuxing Deng, Ganqi Wang, Yongjia Zhang

Abstract

In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity -- a noncollapsed $\mathbb{F}$-limit metric soliton [Bam23,CMZ23]. We first prove some estimates for noncollapsed $\mathbb{F}$-limit metric solitons with nonnegative Ricci curvature, and then obtain two dichotomy theorems for ancient Ricci flows. In particular, we show that: (1) for a noncollapsed ancient Ricci flow with nonnegative Ricci curvature, either its asymptotic volume ratio is always zero, or every tangent flow at infinity is a Ricci flat cone; (2) for a noncollapsed ancient Ricci flow with positively pinched Ricci curvature ($\operatorname{Ric}\ge \varepsilon R g$), either it is compact, or every tangent flow at infinity is a Ricci flat cone.

Ancient Ricci flows with nonnegative Ricci curvature

Abstract

In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity -- a noncollapsed -limit metric soliton [Bam23,CMZ23]. We first prove some estimates for noncollapsed -limit metric solitons with nonnegative Ricci curvature, and then obtain two dichotomy theorems for ancient Ricci flows. In particular, we show that: (1) for a noncollapsed ancient Ricci flow with nonnegative Ricci curvature, either its asymptotic volume ratio is always zero, or every tangent flow at infinity is a Ricci flat cone; (2) for a noncollapsed ancient Ricci flow with positively pinched Ricci curvature (), either it is compact, or every tangent flow at infinity is a Ricci flat cone.

Paper Structure

This paper contains 24 sections, 27 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathbb{F}$-convergence and $\mathbb{F}$-compactness
  4. Convergence of points in $\mathbb{F}$-convergence
  5. Local smooth convergence
  6. Noncollapsed $\mathbb{F}$-limit metric soliton
  7. Basic equations and regularity
  8. The center and the quadratic growth of the potential function
  9. Volume and scalar curvature of sub-level-set
  10. Scalar curvature lower bound
  11. Size of singular set
  12. Curvature and volume estimates of metric solitons
  13. A scalar curvature lower bound
  14. A volume growth bound
  15. AVR of metric solitons
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(X,d,\nu)$ be (the model of) a noncollapsed $\mathbb{F}$-limit metric soliton and $(\mathcal{R}_X,\mathfrak{g},f_0)$ its regular part. Assume that the Ricci curvature is non-negative on $\mathcal{R}_X$ and does not vanish everywhere. Then there exists a positive number $\delta = \delta(X)\in(0,

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 2.1: Bamler's partial regularity result Bam20b
  • Theorem 2.2: Compactness of sequence of points Bam23
  • Theorem 2.3: Bam23 and Bam20b
  • Theorem 2.4: Bam23
  • ...and 35 more