Volume maximization and the extended hyperbolic space
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- by Feng Luo and Jean-Marc Schlenker PDF
- Proc. Amer. Math. Soc. 140 (2012), 1053-1068 Request permission
Abstract:
We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space – the natural extension of hyperbolic space by the de Sitter space – except for the degenerate case where all simplices are Euclidean in a generalized sense.
Those extended hyperbolic structures can realize geometrically a decomposition of the manifold as the connected sum of components admitting a complete hyperbolic metric, along embedded spheres (or projective planes) which are totally geodesic space-like surfaces in the de Sitter part of the extended hyperbolic structure.
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Additional Information
- Feng Luo
- Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
- MR Author ID: 251419
- Jean-Marc Schlenker
- Affiliation: Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse cedex 9, France
- MR Author ID: 362432
- Received by editor(s): June 22, 2010
- Received by editor(s) in revised form: November 27, 2010, and December 8, 2010
- Published electronically: July 1, 2011
- Additional Notes: The first author was partially supported by NSF-DMS0604352.
The second author was partially supported by the ANR program Repsurf: ANR-06-BLAN-0311 - Communicated by: Jianguo Cao
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1053-1068
- MSC (2010): Primary 57M50; Secondary 51M10, 52B70, 57Q15
- DOI: https://doi.org/10.1090/S0002-9939-2011-10941-9
- MathSciNet review: 2869090