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

We study some potential theoretic properties of homothetic solitons Σ n of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in R n + m , we observe similarities and differences in the geometry of solitons in both flows. In particular, we show...

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Veröffentlicht in:The Journal of Geometric Analysis 2021, Vol.31 (1), p.579-618
Hauptverfasser: Gimeno, Vicent, Palmer, Vicente
Format: Artikel
Sprache:eng
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Zusammenfassung:We study some potential theoretic properties of homothetic solitons Σ n of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in R n + m , we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons Σ 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 Σ 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).
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-019-00291-3