Evolution by mean curvature flow in sub-Riemannian geometries:A stochastic approach

Nicolas Dirr; Federica Dragoni; Max von Renesse

doi:10.3934/cpaa.2010.9.307

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2010, Volume 9, Issue 2: 307-326. Doi: 10.3934/cpaa.2010.9.307

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Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach

1.
Department of Mathematical Sciences, University of Bath, Bath, BA1 7AY
2.
Department of Pure Mathematics, Imperial College London, London, SW7 2AZ
3.
Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10632 Berlin

Received: December 31, 2008

Revised: July 31, 2009

Published: February 2010

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Abstract

We study evolution by horizontal mean curvature flow in sub- Riemannian geometries by using stochastic approach to prove the existence of a generalized evolution in these spaces. In particular we show that the value function of suitable family of stochastic control problems solves in the viscosity sense the level set equation for the evolution by horizontal mean curvature flow.

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Mathematics Subject Classification: Primary: 53C44, 53C17, 93E03; Secondary: 35K65, 49L25.

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