Translating solitons of the mean curvature flow asymptotic to hyperplanes in ℝⁿ⁺¹

Eddygledson S. Gama; Francisco Martin

Paper · Feb 23, 2018

Translating solitons of the mean curvature flow asymptotic to hyperplanes in $R^{n + 1}$

Eddygledson S. Gama, Francisco Martin

math.DGWider flowsarXiv:1802.08468v3

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Abstract

A translating soliton is a hypersurface $M$ in $R^{n + 1}$ such that the family $M_{t} = M - t e_{n + 1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point $H = e_{n + 1}^{⊥} .$ In this paper we obtain a characterization of hyperplanes which are parallel to $e_{n + 1}$ and the family of tilted grim reaper cylinders as the only translating solitons in $R^{n + 1}$ which are $C^{1}$ -asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in $R^{3}$ by the second author, Perez-Garcia, Savas-Halilaj and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.

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Submitted: Feb 23, 2018
Last revised: Nov 11, 2018 (v3)
Subjects: math.DG

Comments: 37 pages, 12 figures. This revised version will appear in IMRN

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.

Translating solitons of the mean curvature flow asymptotic to hyperplanes in $R^{n + 1}$

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Translating solitons of the mean curvature flow asymptotic to hyperplanes in Rn+1

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Translating solitons of the mean curvature flow asymptotic to hyperplanes in $R^{n + 1}$