Generalized Alexandrov theorems in spacetimes with integral conditions

Abstract

We investigate integral conditions involving the mean curvature vector $\overrightarrow{H}$ 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 [18] 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.