Perelman-type no breather theorem for noncompact Ricci flows

TL;DR

This paper proves that complete shrinking breathers with bounded Ricci curvature are gradient Ricci solitons, classifies 3D cases, and extends results on Ricci solitons and ancient solutions in Ricci flow.

Contribution

It establishes that certain shrinking breathers are necessarily gradient Ricci solitons and generalizes existing results on Ricci solitons with Ricci curvature bounds.

Findings

Complete shrinking breathers with Ricci curvature bounded below are gradient Ricci solitons.

Classification of all complete 3D shrinking breathers.

Every complete shrinking Ricci soliton with Ricci curvature bounded below is gradient.

Abstract

In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete $3$-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient -- a generalization of Naber's result. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.