Bi-Halfspace and Convex Hull Theorems for Translating Solitons

TL;DR

This work establishes a bi-halfspace constraint for translating solitons of mean curvature flow, proving that a properly immersed self-translater in $\mathbb{R}^{n+1}$ cannot lie in two transverse vertical halfspaces. It then derives a Hoffman–Meeks–style convex-hull classification for the projection $\pi(\Sigma)$, showing $\mathrm{Conv}(\pi(\Sigma))$ must be one of five shapes, with the compact convex-set case precisely when $\Sigma$ is compact; these results are obtained using distance-function methods and the Omori–Yau maximum principle, avoiding barrier constructions. The paper also details the compact and noncompact cases for convex hulls, and provides an Appendix with a 2D alternative proof and discussion of barrier limitations in higher dimensions. Overall, it extends Hoffman–Meeks-type classifications to self-translating solitons, linking projection convex hulls with translator geometry via analytical maximum principles rather than barrier arguments.

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 $Σ^n$ 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 $\mathbb{R}^{n}$.

Figures (1)

• Figure 1:

Theorems & Definitions (49)

• Theorem 1: Bi-Halfspace Theorem
• Theorem 2: Bi-Halfspace Theorem w/ Compact Boundary
• Theorem 3: Convex Hull Classification
• Remark 4
• Remark 5
• Corollary 6
• Remark 7
• Corollary 8: Hoffman-Meeks: hoffman-meeks
• proof : Proof of Corollary \ref{['corr-hm']}
• Corollary 9: Corollary 2.2 xj_wang