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

Abstract

In [ChTi06], Cheeger-Tian proved an ǫ-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 ǫ-regularity theorem which confirms Cheeger-Tian's conjecture with a universal constant ǫ. For Ricci flow, we reduce Cheeger-Tian's ǫ-regularity conjecture to a backward Pseudolocality estimate. By proving a global backward Pseudolocality theorem, we can prove a global ǫ-regularity theorem which partially confirms Cheeger-Tian's conjecture for Ricci flow. Furthermore, as a consequence of the ǫ-regularity, we can show by using the structure theorem of Naber-Tian [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.

Global $\varepsilon$-regularity for 4-dimensional Ricci flow with integral scalar curvature bound

“…In [GJ17], Ge-Jiang proved an ε-regularity for shrinking Ricci solitons which generalizes Cheeger-Tian's result. Moreover, by proving a Backward Pseudolocality estimate for Riemann curvature, Ge-Jiang proved ε-regularity for Ricci flow with bounded scalar curvature, which partially confirms Cheeger-Tian's conjecture in the Ricci flow case.…”

Small curvature concentration and Ricci flow smoothing

“…We do not assume a-priori bounds on the curvature. The novel idea is to obtain local curvature control under the small L n/2 curvature and local entropy bound (see also [21]). We first prove the following result, from which Theorem 1.3 shall follow.…”

The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds

“…We achieve this by estimating separately the traceless Ricci and the Weyl parts of the curvature tensor, using ideas of Haslhoffer-Muller [17] and Donaldson-Sun [14], respectively. After overcoming this difficulty, the proof is fairly standard, and Uhlenbeck's theory [28] of removable singularities along with the ǫ-regularity theorem of [16] let us conclude that in fact the singular GRS has a C ∞ orbifold structure.…”

“…We can now argue as in [7], [26], to conclude that in fact B * has the structure of a C ∞ orbifold at x * . Note that, because we have bounds on f, |∇f | on B * ,the only difference in our setting is that we must use the ǫ-regularity theorem that is Theorem 1.2 of [16]. Also, note that R + |∇f | 2 = f − W(g, f ), |R| ≤ A, and the quadratic growth of f imply that all critical points of f must occur in some bounded set.…”