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arXiv · 2608.13115

Rigidity of stable spacelike capillary hypersurfaces in de Sitter and Minkowski spaces

Hui Ma·Jiaxu Ma·Mingxuan Yang
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Abstract

We prove a rigidity theorem for compact spacelike capillary hypersurfaces in de~Sitter and Minkowski spaces: volume-preserving stability forces total umbilicity when the support is a spacelike totally umbilical hypersurface of nonnegative intrinsic curvature. Using the light-cone model, we construct conformal Killing fields tangent to the support and derive a unified Minkowski-type formula valid in all Lorentzian space forms. The resulting mean-zero functions satisfy an inhomogeneous Jacobi equation and the linearized capillary Robin boundary condition, and form canonical finite-dimensional test families. A finite-trace identity, supplemented in the de~Sitter cases by a nonpositive Dirichlet Green correction, detects the umbilicity defect \(n|h|^2-H^2\) with a definite sign; hence, every non-totally-umbilical hypersurface admits an admissible test function with positive second variation. The construction extends to supports of negative intrinsic curvature, where a unique timelike parameter direction prevents the finite trace from being sign-definite.

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Hui Ma, Jiaxu Ma, Mingxuan Yang. 2026-08-13. Rigidity of stable spacelike capillary hypersurfaces in de Sitter and Minkowski spaces. https://arxiv.org/abs/2608.13115

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