Some topics on Ricci solitons and self-similar solutions to mean curvature flow

TL;DR

This survey explores self-similar solutions to Ricci flow and mean curvature flow, highlighting diameter bounds and extensions of known results to cone manifolds, including special Lagrangians and self-shrinkers.

Contribution

It provides new diameter bounds for Ricci solitons and extends mean curvature flow results from Euclidean spaces to cone manifolds, including toric Calabi-Yau cones.

Findings

Lower diameter bound for shrinking Ricci solitons via Witten-Laplacian eigenvalue estimate

Extension of mean curvature flow results to cone manifolds

Similar diameter bounds for mean curvature flow self-shrinkers

Abstract

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a bound can be obtained from an eigenvalue estimate for a twisted Laplacian, called the Witten-Laplacian. In the second part we discuss self-similar solutions to the mean curvature flow on cone manifolds. Many results have been obtained for solutions in $\bfR^n$ or $\bfC^n$. We see that many of them extend to cone manifolds, and in particular results on $\bfC^n$ for special Lagrangians and self-shrinkers can be extended to toric Calabi-Yau cones. We also see that a similar lower diameter bound can be obtained for self-shrinkers to the mean curvature flow as in the case of shrinking Ricci solitons.