Shadows of graphical mean curvature flow

TL;DR

This paper studies the long-term behavior of mean curvature flow for graphs over domains in R^n with boundary conditions, introducing a novel approach to analyze the flow's projection and boundary mollification.

Contribution

It establishes long-time existence of mean curvature flow for graphs with boundary conditions and introduces a new lemma for boundary mollification respecting curvature.

Findings

Proves long-time existence of the flow under certain conditions.

Analyzes the projection of the flow as a weak solution in R^n.

Provides a new boundary mollification lemma respecting curvature.

Abstract

We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $R^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of the flowing surface onto $R^n$, the shadow of the flow. This moving shadow can be seen as a weak solution for mean curvature flow of hypersurfaces in $R^n$ with a Dirichlet boundary condition. Furthermore, we provide a lemma of independent interest to locally mollify the boundary of an intersection of two smooth open sets in a way that respects curvature conditions.