Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold M²×ℝ

Li Chen; Dan-Dan Hu; Jing Mao; Ni Xiang

Paper · Apr 29, 2018

Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold $M^{2} \times R$

Li Chen, Dan-Dan Hu, Jing Mao, Ni Xiang

math.DGWider flowsarXiv:1804.10864v3

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Abstract

In this paper, for the Lorentz manifold $M^{2} \times R$ , with $M^{2}$ a $2$ -dimensional complete surface with nonnegative Gaussian curvature, we investigate its space-like graphs over compact strictly convex domains in $M^{2}$ , which are evolving by the non-parametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.

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Submitted: Apr 29, 2018
Last revised: May 20, 2018 (v3)
Subjects: math.DG

Comments: 13 papges. Two typos have been corrected in Lemma 3.3 of version 2. Comments are welcome

Mean curvature flow Self-shrinkers & translators Boundaries & non-compact Free boundary & capillary

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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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

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 surfaces of the non-parametric mean curvature flow in Lorentz manifold $M^{2} \times R$

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Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold M2×R

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Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold $M^{2} \times R$