Uniqueness of entire graphs evolving by Mean Curvature flow

TL;DR

This paper establishes the uniqueness of solutions to the graphical mean curvature flow under various conditions, including one-dimensional, rotationally symmetric, and higher-dimensional cases with specific bounds on initial data.

Contribution

It proves the uniqueness of solutions for graphical mean curvature flow in one dimension without extra assumptions and extends these results to higher dimensions with additional conditions.

Findings

Uniqueness in 1D case without assumptions

Extension to rotationally symmetric solutions

Uniqueness in higher dimensions with bounds on initial data

Abstract

In this paper we study the uniqueness of graphical mean curvature flow. We consider as initial conditions graphs of locally Lipschitz functions and prove that in the one dimensional case solutions are unique without any further assumptions. This result is then generalized for rotationally symmetric solutions. In the general $n$- dimensional case, we prove uniqueness under additional conditions: we require a { \em uniform lower bound } on the second fundamental form and the height function of the initial condition. The latter result extends to initial conditions that are proper graphs over subdomains of $\mathbb{R}^n$.