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Convex spacelike hypersurface of constant curvature with boundary on a hyperboloid

Shanze Gao

TL;DR

This work proves a rigidity result for convex spacelike hypersurfaces in Minkowski space with boundary on a hyperboloid or lightcone under constant $k$-th mean curvature and a constant boundary angle. The authors develop integral identities via the Gauss map and an auxiliary potential, then apply the strong maximum principle together with Newton–Maclaurin inequalities to deduce that the surface must be umbilic with $\lambda_i=1$. Consequently, the hypersurface coincides with a portion of a hyperboloid, providing a Minkowski-space analogue of Liebmann-type Euclidean results. The approach highlights how overdetermined boundary data constrains curvature-driven PDEs in Lorentzian geometry, yielding a sharp rigidity statement for spacelike convex hypersurfaces.

Abstract

We consider convex, spacelike hypersurfaces with boundaries on some hyperboloid (or lightcone) in the Minkowski space. If the hypersurface has constant higher order mean curvature, and the angle between the normal vectors of the hypersurface and the hyperboloid (or the lightcone) is constant on the boundary, then the hypersurface must be a part of another hyperboloid.

Convex spacelike hypersurface of constant curvature with boundary on a hyperboloid

TL;DR

This work proves a rigidity result for convex spacelike hypersurfaces in Minkowski space with boundary on a hyperboloid or lightcone under constant -th mean curvature and a constant boundary angle. The authors develop integral identities via the Gauss map and an auxiliary potential, then apply the strong maximum principle together with Newton–Maclaurin inequalities to deduce that the surface must be umbilic with . Consequently, the hypersurface coincides with a portion of a hyperboloid, providing a Minkowski-space analogue of Liebmann-type Euclidean results. The approach highlights how overdetermined boundary data constrains curvature-driven PDEs in Lorentzian geometry, yielding a sharp rigidity statement for spacelike convex hypersurfaces.

Abstract

We consider convex, spacelike hypersurfaces with boundaries on some hyperboloid (or lightcone) in the Minkowski space. If the hypersurface has constant higher order mean curvature, and the angle between the normal vectors of the hypersurface and the hyperboloid (or the lightcone) is constant on the boundary, then the hypersurface must be a part of another hyperboloid.

Paper Structure

This paper contains 8 sections, 5 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vectors in the Minkowski space
  4. Gauss map
  5. Inverse of $dN$
  6. Curvature function
  7. Integral equalities
  8. Proof of Theorem \ref{['thm']}

Key Result

Theorem 1

Suppose $M$ is a convex, spacelike hypersurface in $\mathbb{R}^{n,1}$ with boundary $\partial M$ on a hyperboloid or a lightcone $\widetilde{\Sigma}$. If the $k$-th mean curvature of $M$ is constant, and the angle function $\theta$ is constant, then $M$ must be a part of another hyperboloid.

Figures (1)

  • Figure 1.1: $M$ is on $\widetilde{\Sigma}$.

Theorems & Definitions (8)

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