Generalized Alexandrov theorems in spacetimes with integral conditions

TL;DR

This paper generalizes Alexandrov-type theorems in spacetimes by establishing integral conditions involving mean curvature that determine when a submanifold lies on a shear-free null hypersurface, using weighted Minkowski formulas.

Contribution

It introduces a necessary and sufficient integral curvature condition for submanifolds to be on shear-free null hypersurfaces, relaxing previous curvature restrictions.

Findings

Derived a mean curvature integral inequality criterion.

Extended Alexandrov theorems with weaker curvature assumptions.

Utilized Minkowski formulas with arbitrary weights for rigidity results.

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 $\Sigma$ 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 $\Sigma$ in several aspects. Specifically, we provide a necessary and sufficient condition, in terms of a mean curvature integral inequality, for $\Sigma$ 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.