Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

TL;DR

This paper investigates the rigidity of complete graphical translating hypersurfaces under mean curvature flow when the translation direction is non-graphical, establishing flatness under specific convexity or entropy conditions.

Contribution

It proves flatness of such hypersurfaces when mean convex or entropy is below 2, extending to higher dimensions with growth conditions.

Findings

Graphical translating surfaces are flat if mean convex or entropy < 2.

Higher-dimensional cases also exhibit flatness under growth conditions.

Provides new rigidity results for mean curvature flow in non-graphical directions.

Abstract

In this paper, we study the rigidity results of complete graphical translating hypersurfaces when the translating direction is not in the graphical direction. We proved that any entire graphical translating surface in the translating direction not parallel to the graphical one is flat if either the translating surface is mean convex or the entropy of the translating surface is smaller than $2$. For higher dimensional case, we show that the same conclusion holds if the graphical translating hypersurface satisfies certain growth condition.