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Volume maximization and the extended hyperbolic space


Authors: Feng Luo and Jean-Marc Schlenker
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
Published electronically: July 1, 2011
MathSciNet review: 2869090
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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

Jean-Marc Schlenker
Affiliation: Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse cedex 9, France

DOI: https://doi.org/10.1090/S0002-9939-2011-10941-9
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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