报告人：Nefton Pali （Université Paris-Sud, France）

时间：2015年6月19日（周五）下午 2：00―4:00

地点：管研楼 1418

TITLE 1： The Soliton-Ricci Flow with fixed volume form

Abstract: We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by convexity results for Perelman's W-functional over convex subsets inside adequate subspaces of Riemannian metrics. We show indeed that the Soliton-Ricci flow is generated by the gradient flow of the restriction of Perelman's W-functional over such subspaces. Assuming long time existence of the Soliton-Ricci flow we show exponentially fast convergence to a shrinking Ricci soliton provided that the Bakry-Emery-Ricci tensor is uniformly strictly positive with respect to the evolving metric

TITLE 2：The Soliton-Ricci Flow with variable volume form

Abstract: We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previous work. We still call this new flow the Soliton-Ricci flow. It corresponds to a forward Ricci type flow up to a gauge transformation generated by the gradient of the density of the volumes. The new Soliton-Ricci flow exist for all times and represents the gradient flow of Perelman's W functional with respect to a pseudo-Riemannian structure over the space of metrics and normalized positive volume forms. We obtain an expression of the Hessian of the W functional with respect to such structure. Our expression shows the elliptic nature of this operator in directions orthogonal to the orbits obtained by the action of the group of diffeomorphism. In the case the initial data is K/"ahler then the Soliton-Ricci flow preserves the K/"ahler condition and the symplectic form. The space of tamed complex structures embeds naturally to the space of metrics and normalized positive volume forms via the Chern-Ricci map. Over such space the pseudo-Riemannian structure restricts to a Riemannian one. We perform a study of the sign of the restriction of the Hessian of the W functional over such space. This allows us to obtain a finite dimensional reduction, and thus the solution, of the well known problem of the stability of K/"ahler-Ricci solitons.