ε-regularity for shrinking Ricci solitons and Ricci flows
Abstract
In [Cheeger-Tian 2005], Cheeger-Tian proved an ε-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 ε-regularity theorem which confirms Cheeger-Tian's conjecture with a universal constant ε. For Ricci flow, we reduce Cheeger-Tian's ε-regularity conjecture to a backward Pseudolocality estimate. By proving a global backward Pseudolocality theorem, we can prove a global ε-regularity theorem which partially confirms Cheeger-Tian's conjecture for Ricci flow. Furthermore, as a consequence of the ε-regularity, we can show by using the structure theorem of Naber-Tian NaTi that a collapsed limit of shrinking Ricci solitons with bounded L2 curvature has a smooth Riemannian orbifold structure away from a finite number of points.