Ancient Ricci flows with nonnegative Ricci curvature

TL;DR

This paper investigates the long-term geometric behavior of noncollapsed ancient Ricci flows with nonnegative Ricci curvature, establishing conditions under which tangent flows at infinity are Ricci flat cones or the flow is compact.

Contribution

It proves estimates for noncollapsed $ ext{F}$-limit solitons and establishes dichotomy theorems characterizing the asymptotic geometry of such Ricci flows.

Findings

Either the asymptotic volume ratio is zero or tangent flows at infinity are Ricci flat cones.

For positively pinched Ricci curvature flows, the flow is either compact or tangent flows are Ricci flat cones.

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.