Convexity and the Dirichlet problem of translating mean curvature flows

TL;DR

This paper introduces a new geometric flow called translating mean curvature flow, analyzes its properties, and studies the Dirichlet problem, establishing global existence and convergence results for hypersurface solitons.

Contribution

It proposes a novel translating mean curvature flow and investigates its fundamental properties, including the Dirichlet problem and long-term behavior.

Findings

Positivity preserving property of the flow

Global existence of solutions to the Dirichlet problem

Convergence of the flow over time

Abstract

In this work, we propose a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in $R^{n+1}$. We study the basic properties, such as positivity preserving property, of the translating mean curvature flow. The Dirichlet problem for the graphical translating mean curvature flow is studied and the global existence of the flow and the convergence property are also considered.