Convexity of 2-convex translating solitons to the mean curvature flow in $\mathbb{R}^{n+1}$

TL;DR

This paper proves that complete, uniformly 2-convex translating solitons for the mean curvature flow in Euclidean space are locally strictly convex, leading to the classification of entire graphical solutions as the axisymmetric bowl soliton.

Contribution

It establishes the convexity of uniformly 2-convex translating solitons and classifies entire graphical solutions as the bowl soliton, advancing understanding of mean curvature flow solitons.

Findings

Uniformly 2-convex translating solitons are locally strictly convex.

Entire graphical translating solitons in higher dimensions are the bowl soliton.

The result classifies a broad class of mean curvature flow solutions.

Abstract

We prove that any complete immersed globally orientable uniformly 2-convex translating soliton $\Sigma \subset \mathbb{R}^{n+1}$ for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in $\mathbb{R}^{n+1},\, n\geq 3 $ is the axisymmetric "bowl soliton".