Convexity of 2-convex translating solitons to the mean curvature flow in ℝⁿ⁺¹

Joel Spruck; Liming Sun

Paper · Oct 5, 2019

Convexity of 2-convex translating solitons to the mean curvature flow in $R^{n + 1}$

Joel Spruck, Liming Sun

math.DGWider flowsarXiv:1910.02195v2

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Abstract

We prove that any complete immersed globally orientable uniformly 2-convex translating soliton $Σ \subset R^{n + 1}$ for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in $R^{n + 1}, n \geq 3$ is the axisymmetric "bowl soliton".

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Submitted: Oct 5, 2019
Last revised: May 31, 2020 (v2)
Subjects: math.DG
DOI: 10.1007/s12220-020-00427-w

Comments: To be published on the Journal of Geometric Analysis

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.

Convexity of 2-convex translating solitons to the mean curvature flow in $R^{n + 1}$

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Convexity of 2-convex translating solitons to the mean curvature flow in Rn+1

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Convexity of 2-convex translating solitons to the mean curvature flow in $R^{n + 1}$