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Rigidity of spacelike hypersurface with constant curvature and intersection angle condition

Shanze Gao

TL;DR

The paper proves a rigidity result for spacelike hypersurfaces in Minkowski space with boundary on a spacelike hyperplane: if the hypersurface has constant $k$-th mean curvature $H_k$ and its boundary meets the hyperplane at a constant angle, then the hypersurface is either contained in the hyperplane or is a portion of a hyperboloid. The authors reduce to a standard position via an isometry, establish $k$-convexity, derive a Serrin-type integral identity, and employ an auxiliary function $P$ together with a maximum principle to deduce total umbilicity. Consequently, the only nontrivial example is a hyperboloid given (in explicit form) by $u=c+\theta_0+\sqrt{1+|x-a|^2}$ on a Euclidean ball, providing a Minkowski-space analogue of the Euclidean sphere theorem under an intersection-angle boundary condition. The result highlights how overdetermined boundary data rigidity in Lorentzian geometry leads to highly symmetric, canonical hypersurfaces.

Abstract

In the Minkowski space, we consider a spacelike hypersurface with boundary, which can be written as a graph on a spacelike hyperplane. We prove that, if its $k$-th mean curvature is constant, and its boundary is on the hyperplane with constant intersection angles, then the hypersurface must be a part of a hyperboloid, unless it is entirely contained in the hyperplane. This result can be seen as an analog of the sphere theorem for the hypersurface of constant $k$-th mean curvature in the Euclidean space.

Rigidity of spacelike hypersurface with constant curvature and intersection angle condition

TL;DR

The paper proves a rigidity result for spacelike hypersurfaces in Minkowski space with boundary on a spacelike hyperplane: if the hypersurface has constant -th mean curvature and its boundary meets the hyperplane at a constant angle, then the hypersurface is either contained in the hyperplane or is a portion of a hyperboloid. The authors reduce to a standard position via an isometry, establish -convexity, derive a Serrin-type integral identity, and employ an auxiliary function together with a maximum principle to deduce total umbilicity. Consequently, the only nontrivial example is a hyperboloid given (in explicit form) by on a Euclidean ball, providing a Minkowski-space analogue of the Euclidean sphere theorem under an intersection-angle boundary condition. The result highlights how overdetermined boundary data rigidity in Lorentzian geometry leads to highly symmetric, canonical hypersurfaces.

Abstract

In the Minkowski space, we consider a spacelike hypersurface with boundary, which can be written as a graph on a spacelike hyperplane. We prove that, if its -th mean curvature is constant, and its boundary is on the hyperplane with constant intersection angles, then the hypersurface must be a part of a hyperboloid, unless it is entirely contained in the hyperplane. This result can be seen as an analog of the sphere theorem for the hypersurface of constant -th mean curvature in the Euclidean space.

Paper Structure

This paper contains 7 sections, 4 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minkowski space
  4. Hypersurface in $\mathbb{R}^{n,1}$
  5. Symmetric functions of curvature
  6. The $k$-convexity of the hypersurface
  7. Proof of the theorem

Key Result

Theorem 1

Suppose $M$ is a connected, spacelike hypersurface with boundary $\partial M$ in $\mathbb{R}^{n,1}$, which is a graph on a spacelike hyperplane $P$. If the $k$-th mean curvature of $M$ is constant, and $\partial M$ is on $P$ with constant intersection angles, then $M$ is either a part of a hyperbolo

Theorems & Definitions (8)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Proposition 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof