Communications in Analysis and Geometry

Volume 19 (2011)

Number 5

On Type-I singularities in Ricci flow

Pages: 905 – 922

DOI: http://dx.doi.org/10.4310/CAG.2011.v19.n5.a4

Authors

Joerg Enders (Institut für Mathematik, Universität Potsdam, Germany)

Reto Müller (Department of Mathematics, Imperial College London, United Kingdom)

Peter M. Topping (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Abstract

We define several notions of singular set for Type-I Ricci flows andshow that they all coincide. In order to do this, we prove thatblow-ups around singular points converge to nontrivialgradient shrinking solitons, thus extending work of Naber.As a by-product we conclude that the volume of afinite-volume singular set vanishes at the singular time.

We also define a notion of density for Type-I Ricci flows and use itto prove a regularity theorem reminiscent of White’s partialregularity result for mean curvature flow.

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