Translating solitons of the mean curvature flow asymptotic to hyperplanes in $\mathbb{R}^{n+1}$

TL;DR

This paper characterizes certain translating solitons in Euclidean space that are asymptotic to half-hyperplanes, identifying hyperplanes and tilted grim reaper cylinders as the only such solutions under specific asymptotic conditions.

Contribution

It extends previous results by classifying all $C^1$-asymptotic translating solitons in higher dimensions, removing previous restrictions on genus and cylinder orientation.

Findings

Hyperplanes parallel to $ extbf{e}_{n+1}$ are characterized as translating solitons.

Tilted grim reaper cylinders are identified as the only other asymptotic solutions.

The classification holds for $C^1$-asymptotic solitons outside a non-vertical cylinder.

Abstract

A translating soliton is a hypersurface $M$ in $\mathbb{R}^{n+1}$ such that the family $M_t= M- t \,\mathbf{e}_{n+1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf{e}_{n+1}^{\perp}.$ In this paper we obtain a characterization of hyperplanes which are parallel to $\mathbf{e}_{n+1}$ and the family of tilted grim reaper cylinders as the only translating solitons in $\mathbb{R}^{n+1}$ which are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in $\mathbb{R}^3$ by the second author, Perez-Garcia, Savas-Halilaj and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.