Parabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow

TL;DR

This paper investigates the geometric and potential theoretic properties of self-similar solutions to the mean curvature flow and inverse mean curvature flow, revealing their parabolicity, properness, and classification under certain conditions.

Contribution

It provides new insights into the geometry of solitons for MCF and IMCF, including their classification, parabolicity, and behavior in bounded regions, using extrinsic distance and Brownian motion analysis.

Findings

Parabolic MCF-solitons with n>2 are self-shrinkers.

Parabolic IMCF-solitons are self-expanders.

Properly immersed MCF-self-shrinkers with bounded second fundamental form are classified.

Abstract

We study some potential theoretic properties of homothetic solitons $\Sigma^n$ of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in $\mathbb{R}^{n+m}$, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons $\Sigma^n$ with $n>2$ are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the Mean Exit Time function for the Brownian motion defined on $\Sigma$ as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of \cite{CaoLi}.