德西特空间与闵可夫斯基空间中稳定类时毛细管超曲面的刚性
Rigidity of stable spacelike capillary hypersurfaces in de Sitter and Minkowski spaces
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中文总结 AI 辅助
该研究证明了德西特与闵可夫斯基空间中，当支撑为非负内蕴曲率的类时全脐超曲面时，保体积稳定的紧致类时毛细管超曲面必为全脐，还推导了统一闵可夫斯基型公式并构造了测试族。
中文摘要 AI 辅助
我们证明了德西特空间与闵可夫斯基空间中紧致类时毛细管超曲面的一个刚性定理：当支撑是具有非负内蕴曲率的类时全脐超曲面时，保体积稳定性迫使该超曲面成为全脐的。利用光锥模型，我们构造了与支撑相切的共形基灵场，并推导了适用于所有洛伦兹空间形式的统一闵可夫斯基型公式。所得的均值为零的函数满足非齐次雅可比方程和线性化毛细管罗宾边界条件，构成了典型的有限维测试族。有限迹恒等式在德西特情形下由非正狄利克雷格林修正项补充，可检测脐性缺陷\ni|h|² - H² 并得到确定符号；因此，每个非全脐超曲面都存在具有正二阶变分的容许测试函数。该构造可扩展到具有负内蕴曲率的支撑，其中唯一的类时参数方向使有限迹不具有确定符号。
英文摘要
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.
