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arXiv · 2307.09287

Generalized Alexandrov theorems in spacetimes with integral conditions

Kwok-Kun Kwong·Xianfeng Wang
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Abstract

We investigate integral conditions involving the mean curvature vector $\vec{H}$ or mixed higher-order mean curvatures, to determine when a codimension-two submanifold $Σ$ lies on a shear-free (umbilical) null hypersurface in a spacetime. We generalize the Alexandrov-type theorems in spacetime introduced in \cite{wang2017Minkowski} by relaxing the curvature conditions on $Σ$ in several aspects. Specifically, we provide a necessary and sufficient condition, in terms of a mean curvature integral inequality, for $Σ$ to lie in a shear-free null hypersurface. A key component of our approach is the use of Minkowski formulas with arbitrary weight, which enables us to derive rigidity results for submanifolds with significantly weaker integral curvature conditions.

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Kwok-Kun Kwong, Xianfeng Wang. 2023-07-18. Generalized Alexandrov theorems in spacetimes with integral conditions. https://arxiv.org/abs/2307.09287

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