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).

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Correspondence to Vicente Palmer.

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Vicent Gimeno: Work partially supported by the Research Program of University Jaume I Project UJI-B2018-35, and DGI -MINECO Grant (FEDER) MTM2017-84851-C2-2-P. Vicente Palmer: Work partially supported by the Research Program of University Jaume I Project UJI-B2018-35, DGI-MINECO Grant (FEDER) MTM2017-84851-C2-2-P, and Generalitat Valenciana Grant PrometeoII/2014/064.

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Gimeno, V., Palmer, V. Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow. J Geom Anal 31, 579–618 (2021). https://doi.org/10.1007/s12220-019-00291-3

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Keywords

  • Extrinsic distance
  • Parabolicity
  • Soliton
  • Self-shrinker
  • Self-expander
  • Mean exit time function
  • Laplace operator
  • Brownian motion
  • Mean curvature flow
  • Inverse mean curvature flow

Mathematics Subject Classification

  • Primary 53C21
  • 53C44
  • Secondary 53C42
  • 58J65
  • 60J65