Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds

TL;DR

This paper establishes a priori estimates and existence results for the mean curvature flow of graphs in Cartan-Hadamard manifolds with asymptotic boundary conditions, using convexity at infinity to ensure regularity.

Contribution

It introduces a new approach to analyze mean curvature flow with asymptotic Dirichlet conditions in non-compact manifolds, providing existence and regularity results.

Findings

A priori estimates for mean curvature flow in Cartan-Hadamard manifolds.

Existence of solutions with asymptotic Dirichlet boundary conditions.

Construction of barriers ensuring regularity at infinity.

Abstract

A priori estimates for the mean curvature evolution of Killing graphs in Cartan-Hadamard manifolds with asymptotic Dirichlet conditions are established. As an application, the existence of the corresponding parabolic flow is proved, ensuring regularity of the obtained solutions through the construction of suitable barriers at points of the asymptotic boundary. Such a construction is made possible under an appropriate notion of convexity at infinity.