On Type-I singularities in Ricci flow

Abstract

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber [15]. As a byproduct we conclude that the volume of a finite-volume singular set vanishes at the singular time.We also define a notion of density for Type I Ricci flows and use it to prove a regularity theorem reminiscent of White's partial regularity result for mean curvature flow [22].

Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers

“…A consequence of Theorem 1.5 is the following corollary whose analog has been proved for the type-I Ricci flow in [8]. For the case of the mean curvature flow with H ≥ −C, having type-I singularities, we can prove a stronger statement.…”

“…For type-I Ricci flow, Enders et al [8] obtained a lower bound on the blow-up rate for the scalar curvature at the first singular time of the Ricci flow, similar to (1.5). Their result and ours have been proved by blow up arguments.…”

“…An analogous statement for the type-I Ricci flow has been obtained in [8]. Our goal in this paper is to show that Σ H = Σ, that is all notions of singular sets coincide for any type-I mean curvature flow, without requiring the mean convexity.…”

“…In [8], Ender, Müller and Topping established the blow-up rate of the scalar curvature at any singular point of type-I Ricci flow. In [19], Stone established the blow-up rate of the second fundamental form and the mean curvature at any singular point of the mean convex mean curvature flow having type-I singularities.…”

“…This is a kind of a rigidity result for self-shrinkers with nonnegative mean curvature. Note that, in [8], in order to establish the blow-up rate of the scalar curvature at any singular point of type-I Ricci flow, the following rigidity result for gradient shrinking solitons, due to Pigola-Rimoldi-Setti, played an important role:…”