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 |
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Hauptverfasser: | , |
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). |
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ISSN: | 1050-6926 1559-002X |
DOI: | 10.1007/s12220-019-00291-3 |