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

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 نشر من قبل Vicente Palmer
 تاريخ النشر 2018
  مجال البحث
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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}.



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