A remark on gradient estimates for spacelike mean curvature flow with boundary conditions

TL;DR

This paper establishes a gradient estimate for spacelike mean curvature flow with boundary conditions in two dimensions, ensuring long-term existence and convergence to a translating solution.

Contribution

It provides the first gradient estimate for this flow with general Neumann boundary conditions in dimension two.

Findings

Flow exists for all time under the given conditions.

Flow converges to a translating solution.

Gradient estimates are achieved for the first time in this setting.

Abstract

We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension $n=2$. This then implies that the mean curvature flow exists for all time and converges to a translating solution.