Calculus of Variations and Geometric Measure Theory

A density function for the Ricci Flow

Carlo Mantegazza (Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università di Napoli Federico II and Scuola Superiore Meridionale, Napoli, Italy)

created by magnani on 22 Mar 2012

28 mar 2012 -- 17:00   [open in google calendar]

Sala Seminari, Department of Mathematics, Pisa University

Abstract.

Inspired by the density function for mean curvature flow introduced by A. Stone and based on Huisken's monotonicity formula, we define a similar quantity for the Ricci flow, related to the monotonicity of Perelman's W-functional. We discuss its connections with the singularities in the "Type-I case" and we give an alternative proof of the result of Enders, Mueller and Topping that around a type-one singular point, blowing-up the flow in a suitable way, one obtains a shrinking gradient Ricci soliton in the geometric limit (in every dimension). In perspective, this line of analysis could work also for general singular points (also type-II) in dimension two and three (and very hopefully four). This would give another and more natural method to get an asymptotic shrinking gradient Ricci soliton, alternative to Perelman's blow-upblow-down procedure.