The Björling problem for prescribed mean curvature surfaces in ℝ³

Antonio Bueno

Paper · May 6, 2018

The Björling problem for prescribed mean curvature surfaces in $R^{3}$

Antonio Bueno

math.DGWider flowsarXiv:1805.02251v2

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Abstract

In this paper we solve the Björling problem for the class of immersed surfaces in $R^{3}$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Möbius strip for an arbitrary large class of prescribed functions. In particular, we use the Björling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Möbius strip in $R^{3}$

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Submitted: May 6, 2018
Last revised: Mar 18, 2019 (v2)
Subjects: math.DG

Comments: 6 figures

Mean curvature flow Self-shrinkers & translators

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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)

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 Björling problem for prescribed mean curvature surfaces in $R^{3}$

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The Björling problem for prescribed mean curvature surfaces in R3

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The Björling problem for prescribed mean curvature surfaces in $R^{3}$