Complete translating solitons to the mean curvature flow in $\mathbb{R}^3$ with nonnegative mean curvature

TL;DR

This paper proves that complete, mean convex translating solitons in three-dimensional space are convex, classifies entire graphical solutions as the bowl soliton, and describes all convex graphical solitons over strips.

Contribution

It establishes convexity of complete mean convex translating solitons in or the first time and classifies entire and strip-region solutions.

Findings

Complete mean convex translating solitons are convex.

Entire mean convex graphical solitons are the bowl soliton.

Classifies convex graphical solitons over strip regions.

Abstract

We prove that any complete immersed two-sided mean convex translating soliton $\Sigma \subset \mathbb{R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of $\Sigma$ tends to zero at infinity, then $\Sigma$ can be represented as an entire graph and so is the "bowl soliton". Finally we classify all locally strictly convex graphical translating solitons defined over strip regions.