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

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.

References in corpus (3)

• On complete embedded translating solitons of the mean curvature flow that are of finite genus
• Finite topology self-translating surfaces for the mean curvature flow in ${\mathbb{R}}^3$
• The extension problem of the mean curvature flow (I)

Cited by in corpus (7)

• Bi-Halfspace and Convex Hull Theorems for Translating Solitons
• The global geometry of surfaces with prescribed mean curvature in ${\mathbb{R}}^3$
• Convexity of 2-convex translating solitons to the mean curvature flow in ${\mathbb{R}}^{n+1}$
• Translating solitons of the mean curvature flow in the space ${\mathbb{H}}^2\times \mathbb{R}$
• Generalized solutions to the Dirichlet problem of translating mean curvature equations
• Graphical translating solitons for the mean curvature flow and isoparametric functions
• Flowing the leaves of a foliation with normal speed given by the logarithm of general curvature functions