We show that recent work of Song implies that torsion‐free hyperbolic groups with Gromov boundary arerealized as fundamental groups of closed 3‐manifolds of constant negative curvature if and only if the solution to an associated spherical Plateau problem for group homology is isometric to such a 3‐manifold, and suggest some related questions.
Simple numerical solutions to the Einstein constraints on various three-manifolds
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.
more » « less- Award ID(s):
- 2012857
- NSF-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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