Bi-Halfspace and Convex Hull Theorems for Translating Solitons. (8th October 2019)

• Record Type: Journal Article
• Authors: Chini, Francesco; Møller, Niels Martin
• Is Part Of: International mathematics research notices. Volume 2021:Number 17(2021)
• Page Start: 13011
• Page End: 13045
• Publication Date: 2019-10-08
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnz183
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: While it is well known from examples that no interesting "halfspace theorem" holds for properly immersed $n$ -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" (aka "wedge theorem"): two transverse vertical halfspaces can never contain the same such hypersurface. The same holds for any infinite end. The proofs avoid the typical methods of nonlinear barrier construction for the approach via distance functions and the Omori–Yau maximum principle. As an application we classify the closed convex hulls of all properly immersed (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 ${\mathbb {R}}^{n}$ .