Minkowski formulae and Alexandrov theorems in spacetime

Mu-Tao Wang Columbia University Ye-Kai Wang Columbia University Xiangwen Zhang University of California, Irvine

Differential Geometry Geometric Analysis and Geometric Topology Mathematical Physics mathscidoc:1608.10014

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.
Minkowski formulae, Alexandrov theorems
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  title={Minkowski formulae and Alexandrov theorems in spacetime},
  author={Mu-Tao Wang, Ye-Kai Wang, and Xiangwen Zhang},
Mu-Tao Wang, Ye-Kai Wang, and Xiangwen Zhang. Minkowski formulae and Alexandrov theorems in spacetime. 2014. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160820133107813504318.
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