Paper · Sep 4, 2018

Bi-Halfspace and Convex Hull Theorems for Translating Solitons

Francesco Chini, Niels Martin Møller

math.DGWider flowsarXiv:1809.01069v2
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Abstract

While it is well known from examples that no interesting `halfspace theorem' holds for properly immersed complete -dimensional self-translating mean curvature flow solitons in Euclidean space , 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) -dimensional mean curvature flow self-translating solitons in , 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 , halfspaces, slabs, hyperplanes and convex compacts in .

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Submitted
Sep 4, 2018
Last revised
Mar 6, 2019 (v2)
Subjects
math.DG, math.AP
Journal
Int. Math. Res. Not. IMRN 2021, no. 17, 13011-13045. MR4307681
DOI
10.1093/imrn/rnz183

Comments: 30 pages; Minor edits and added references

Mean curvature flowSelf-shrinkers & translatorsMinimal & CMC surfaces

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.