On Type-I singularities in Ricci flow

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 by-product 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].