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

TL;DR

This paper proves that complete 2-convex translating and expanding solitons to the mean curvature flow in Euclidean space are convex, extending understanding of their geometric properties and convexity conditions.

Contribution

It establishes the convexity of complete 2-convex translating and expanding solitons in Euclidean space, generalizing previous results and clarifying their geometric structure.

Findings

Complete 2-convex translating solitons are convex.

Complete 2-convex self-expanders asymptotic to mean convex cones are convex.

Results extend convexity properties to higher dimensions for these solitons.

Abstract

In this paper, inspired by the work of Spruck-Xiao [27] and based partly on a result of Derdzi\'nski [11], we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in $\mathbb{R}^{n+1}$. More precisely, for $n\geq 3$, we show that any $n$-dimensional complete 2-convex translating solitons are convex, and any $n$-dimensional complete 2-convex self-expanders asymptotic to (strictly) mean convex cones are convex.