Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold $M^{2}\times\mathbb{R}$

TL;DR

This paper studies the evolution of space-like graphs in a Lorentz manifold under non-parametric mean curvature flow with boundary conditions, showing solutions tend to translate over time.

Contribution

It introduces a new analysis of non-parametric mean curvature flow in Lorentz manifolds with convex boundary conditions, demonstrating convergence to translating solutions.

Findings

Solutions converge to translating solutions over time

Flow preserves the space-like condition and boundary contact angle

Provides new insights into geometric evolution in Lorentzian settings

Abstract

In this paper, for the Lorentz manifold $M^{2}\times\mathbb{R}$, with $M^{2}$ a $2$-dimensional complete surface with nonnegative Gaussian curvature, we investigate its space-like graphs over compact strictly convex domains in $M^{2}$, which are evolving by the non-parametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.