$\epsilon$-regularity for shrinking Ricci solitons and Ricci flows

TL;DR

This paper advances understanding of regularity in Ricci flow and solitons by proving new epsilon-regularity results, constructing counterexamples, and linking geometric limits to orbifold structures.

Contribution

It proves epsilon-regularity for 4D shrinking Ricci solitons, constructs counterexamples for certain conjectures, and establishes a global pseudolocality theorem for Ricci flows.

Findings

Confirmed epsilon-regularity for 4D shrinking Ricci solitons.

Reduced Ricci flow epsilon-regularity to a pseudolocality estimate.

Showed collapsed limits of Ricci solitons have orbifold structures.

Abstract

In [Cheeger-Tian 2005], Cheeger-Tian proved an $\epsilon$-regularity theorem for $4$-dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar curvature, shrinking Ricci solitons, Ricci flows in $4$-dimensional manifolds and higher dimensional Einstein manifolds. In this paper we consider all these problems. First, we construct counterexamples to the conjecture for $4$-dimensional critical metrics and counterexamples to the conjecture for higher dimensional Einstein manifolds. For $4$-dimensional shrinking Ricci solitons, we prove an $\epsilon$-regularity theorem which confirms Cheeger-Tian's conjecture with a universal constant $\epsilon$. For Ricci flow, we reduce Cheeger-Tian's $\epsilon$-regularity conjecture to a backward Pseudolocality estimate. By proving a global backward Pseudolocality theorem, we can prove a global $\epsilon$-regularity theorem which partially confirms Cheeger-Tian's conjecture for Ricci flow. Furthermore, as a consequence of the $\epsilon$-regularity, we can show by using the structure theorem of Naber-Tian \cite{NaTi} that a collapsed limit of shrinking Ricci solitons with bounded $L^2$ curvature has a smooth Riemannian orbifold structure away from a finite number of points.