Translating graphs by mean curvature flow

TL;DR

This paper classifies and analyzes translating graphs under mean curvature flow in three-dimensional space, establishing their types, stability, and curvature bounds, and proving non-existence over bounded domains.

Contribution

It provides a complete classification of translating graphs in b2^3, proves their stability, and establishes curvature bounds, advancing understanding of mean curvature flow solutions.

Findings

Non-existence of complete translating graphs over bounded domains in b2^2.

Classification of complete translating graphs into three types.

Proof of stability and curvature bounds for these graphs.

Abstract

The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of complete translating graphs in $\Real^3$; entire graphs, graphs between two vertical planes, and graphs in one side of a plane. In the last two types, graphs are asymptotic to planes next to their boundaries. We also prove stability of translating graphs and then we obtain a pointwise curvature bound for translating graphs in $\Real^3$.