An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ℝⁿ⁺¹₁

Ya Gao; Jing Mao

Paper · Jun 10, 2021

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space $R_{1}^{n + 1}$

Ya Gao, Jing Mao

math.DGWider flowsarXiv:2106.05973

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Abstract

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $H^{n} (1)$ , of center at origin and radius $1$ , in the $(n + 1)$ -dimensional Lorentz-Minkowski space $R_{1}^{n + 1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of hyperbolic plane of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of $H^{n} (1)$ , as time tends to infinity. Clearly, this conclusion is an extension of our previous work [2].

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Submitted: Jun 10, 2021
Subjects: math.DG

Comments: 16 pages. Our related works (i.e., references [3,4,5] here) will also be put on arXiv soon. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2104.10600, arXiv:2104.08884

Mean curvature flow Other curvature flows Inverse mean curvature flow Boundaries & non-compact Free boundary & capillary Long-time existence & convergence Physics

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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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 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)

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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.

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space $R_{1}^{n + 1}$

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An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space R1n+1​

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An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space $R_{1}^{n + 1}$