The global geometry of surfaces with prescribed mean curvature in ℝ³

Antonio Bueno; Jose A. Galvez; Pablo Mira

Paper · Feb 22, 2018

The global geometry of surfaces with prescribed mean curvature in $R^{3}$

Antonio Bueno, Jose A. Galvez, Pablo Mira

math.DGWider flowsarXiv:1802.08146v2

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Abstract

We develop a global theory for complete hypersurfaces in $R^{n + 1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in $R^{n + 1}$ , and also that of self-translating solitons of the mean curvature flow. For the particular case $n = 2$ , we will obtain results regarding a priori height and curvature estimates, non-existence of complete stable surfaces, and classification of properly embedded surfaces with at most one end.

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Submitted: Feb 22, 2018
Last revised: Feb 25, 2019 (v2)
Subjects: math.DG

Comments: 35 pages. We have substantially shortened the paper with respect to version 1, following the editor's suggestion

Mean curvature flow Self-shrinkers & translators Minimal & CMC surfaces

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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

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.

[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.

The global geometry of surfaces with prescribed mean curvature in $R^{3}$

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The global geometry of surfaces with prescribed mean curvature in R3

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The global geometry of surfaces with prescribed mean curvature in $R^{3}$