Calculus of Variations and Geometric Measure Theory

A density function for the Ricci Flow

Carlo Mantegazza (Dip. Mat. "Renato Caccioppoli", Univ. Napoli Federico II and Scuola Superiore Meridionale, Napoli)

created by magnani on 22 Mar 2012

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

Sala Seminari, Department of Mathematics, Pisa University


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.