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Abstract We prove the rigidity ofrectifiableboundaries with constantdistributionalmean curvature in the Brendle class of warped product manifolds (which includes important models in general relativity, like the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds).As a corollary, we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant.The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong C^{k}-norms.Our method also establishes that rectifiable boundaries of sets of finite perimeter in the hyperbolic space with constant distributional mean curvature are finite unions of possibly mutually tangent geodesic spheres.
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Simple numerical solutions to the Einstein constraints on various three-manifolds
Abstract Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature.
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- Award ID(s):
- 2012857
- PAR ID:
- 10377017
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- General Relativity and Gravitation
- Volume:
- 54
- Issue:
- 10
- ISSN:
- 0001-7701
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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