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

Pages: 307 - 326, Volume 9, Issue 2, March 2010

doi:10.3934/cpaa.2010.9.307       Abstract        Full Text (265.7K)       Related Articles

Nicolas Dirr - Department of Mathematical Sciences, University of Bath, Bath, BA1 7AY, United Kingdom (email)
Federica Dragoni - Department of Pure Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom (email)
Max von Renesse - Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10632 Berlin, Germany (email)

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.

Keywords:  Mean curvature flow, sub-Riemannian geometries, level set equation, stochastic processes and control.
Mathematics Subject Classification:  Primary: 53C44, 53C17, 93E03; Secondary: 35K65, 49L25.

Received: January 2009;      Revised: August 2009;      Published: December 2009.