Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow

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 Cao and Li (Calc Var 46:879–889, 2013).