While it is well known from examples that no interesting "halfspace theorem" holds for properly immersed n-dimensional selftranslating mean curvature flow solitons in Euclidean space 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 convex hulls of all properly immersed (possibly with compact boundary) n-dimensional mean curvature flow self-translating solitons Σ n in 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 R n , halfspaces, slabs, hyperplanes and convex compacts in R n .
In this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ℝ3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by Spruck and Xiao are convex. Recently, such 2-dimensional convex translating solitons have been completely classified, up to an ambient isometry, as vertical planes, (tilted) grim reaper cylinders, Δ-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by Ho man, Martín and White show that the bound on the entropy is necessary.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.