$${\epsilon}$$ ϵ -regularity for shrinking Ricci solitons and Ricci flows

Abstract

Cheeger and Tian (J Am Math Soc 19(2):487–525, 2006) 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 and Tian (Geometric structures of collapsing Riemannian manifolds I. Surveys in geometric analysis and relativity, International Press, Somerville, 2011) 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.