Mean curvature flow of graphs with asymptotic dirichlet conditions in cartan-hadamard manifolds

Abstract

This work approach the mean curvature evolution of Killing graphs in Cartan-Hadamard manifolds with asymptotic Dirichlet conditions. In order to proof the existence of the flow, a priori estimates are obtained, which ensure the use of the theory of parabolic partial differential equations. The regularity of the obtained solution is studied, building barriers at the points of the asymptotic frontier. Such a construction is possible when considering a concept of convexity at infinity. This thesis also deals with the more general problem of the evolution of graphs by a function of their principal curvatures. In this case, under some conditions, an a priori (interior) gradient estimate is obtained.