Bi-Halfspace and Convex Hull Theorems for Translating Solitons

Abstract

While it is well known from examples that no interesting `halfspace theorem' holds for properly immersed complete $n$-dimensional self-translating mean curvature flow solitons in Euclidean space ${\mathbb{R}}^{n+1}$, we show that they must all obey a general `bi-halfspace theorem': Two transverse vertical halfspaces can never contain the same such hypersurface. The proof avoids the typical methods of nonlinear barrier construction, not readily available here, for the approach via distance functions and the Omori-Yau maximum principle. As an application we classify the convex hulls of all properly immersed complete (possibly with compact boundary) $n$-dimensional mean curvature flow self-translating solitons ${\Sigma }^n$ in ${\mathbb{R}}^{n+1}$, up to an orthogonal projection in the direction of translation. This list is short, coinciding with the one given by Hoffman-Meeks in 1989, for minimal submanifolds: All of ${\mathbb{R}}^n$, halfspaces, slabs, hyperplanes and convex compacts in ${\mathbb{R}}^n$.

30 pages; Minor edits and added references

References in corpus (6)

• Evolution of curves and surfaces by mean curvature
• Complete translating solitons to the mean curvature flow in ${\mathbb{R}}^3$ with nonnegative mean curvature
• Non-existence for self-translating solitons
• Some results on translating solitons of the mean curvature flow
• Ancient mean curvature flows and their spacetime tracks
• Translating solitons of the mean curvature flow in the space ${\mathbb{H}}^2\times \mathbb{R}$

Cited by in corpus (3)

• Ancient mean curvature flows and their spacetime tracks
• The atomic structure of ancient grain boundaries
• Jenkins-Serrin problem for translating horizontal graphs in $M\times \mathbb{R}$