Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ${\mathbb{R}}_1^{n+1}$

Abstract

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane ${\mathcal{H}}^n(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space ${\mathbb{R}}_1^{n+1}$ along the 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 ${\mathcal{H}}^n(1)$, as time tends to infinity.

arXiv comment21 pages. Several minor revisions, two examples (see Section 7) and 5 references have been added to v4. Comments are welcome

References in corpus (5)

• A geometric inequality on hypersurface in hyperbolic space
• Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold
• Inverse mean curvature flow inside a cone in warped products
• A stability result for translating space-like graphs in Lorentz manifolds
• A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone

Cited by in corpus (3)

• An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ${\mathbb{R}}_1^{n+1}$
• An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane ${\mathbb{R}}_1^2$
• Curvature estimates for spacelike graphic hypersurfaces in Lorentz-Minkowski space ${\mathbb{R}}_1^{n+1}$