TY - JOUR

T1 - Minkowski formulae and Alexandrov theorems in space time

AU - Wang, Mu Tao

AU - Wang, Ye Kai

AU - Zhang, Xiangwen

N1 - Funding Information:
M.-T. Wang is supported by NSF grant DMS-1105483 and DMS-1405152. X.W. Zhang is supported by NSF grant DMS-1308136. This work was partially supported by a grant from the Simons Foundation (#305519 to Mu-Tao Wang)

PY - 2017/2

Y1 - 2017/2

N2 - 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.

AB - 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.

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U2 - 10.4310/jdg/1486522815

DO - 10.4310/jdg/1486522815

M3 - Article

AN - SCOPUS:85017470070

VL - 105

SP - 249

EP - 290

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 2

ER -