Ricci-mean curvature flows in gradient shrinking Ricci solitons

It was proved by Huisken that a mean curvature flow converges to a self-shrinker in the Euclidean space after scaling when it develops a singularity of type I. In this paper, we study a coupled flow with a mean curvature flow and a Ricci flow, and generalize his result for this Ricci-mean curvature flow. Then, as a parabolic rescaling limit, we get a self-shrinker in a gradient shrinking Ricci soliton in the sense of Lott under some assumptions.

Keywords

mean curvature flow, Ricci flow, self-similar solution, gradient soliton

2010 Mathematics Subject Classification

53C42, 53C44

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This work was supported by Grant-in-Aid for JSPS Fellows Grant Number 13J06407, and by the Program for Leading Graduate Schools, MEXT, Japan.

Received 31 December 2016

Accepted 18 April 2019

Published 21 August 2020