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

Published electronically:
July 1, 2011

MathSciNet review:
2869090

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Abstract | References | Similar Articles | Additional Information

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.

**[CK06]**Yunhi Cho and Hyuk Kim,*Volume of 𝐶^{1,𝛼}-boundary domain in extended hyperbolic space*, J. Korean Math. Soc.**43**(2006), no. 6, 1143–1158. MR**2264675**, 10.4134/JKMS.2006.43.6.1143**[KL]**A. Kitaev and Feng Luo.

Existence of angle structures on 3-manifolds,

in preparation.**[Luo93]**Feng Luo,*Triangulations in Möbius geometry*, Trans. Amer. Math. Soc.**337**(1993), no. 1, 181–193. MR**1123456**, 10.1090/S0002-9947-1993-1123456-9**[Luo07]**Feng Luo,*Volume and angle structures on 3-manifolds*, Asian J. Math.**11**(2007), no. 4, 555–566. MR**2402938**, 10.4310/AJM.2007.v11.n4.a2**[Mil94]**John Milnor,*Collected papers. Vol. 1*, Publish or Perish, Inc., Houston, TX, 1994. Geometry. MR**1277810****[Sch98]**Jean-Marc Schlenker,*Métriques sur les polyèdres hyperboliques convexes*, J. Differential Geom.**48**(1998), no. 2, 323–405 (French, with English and French summaries). MR**1630178****[Sch01]**Jean-Marc Schlenker,*Convex polyhedra in Lorentzian space-forms*, Asian J. Math.**5**(2001), no. 2, 327–363 (English, with English and French summaries). MR**1868937**, 10.4310/AJM.2001.v5.n2.a3**[Vin93]***Geometry. II*, Encyclopaedia of Mathematical Sciences, vol. 29, Springer-Verlag, Berlin, 1993. Spaces of constant curvature; A translation of Geometriya. II, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988; Translation by V. Minachin [V. V. Minakhin]; Translation edited by È. B. Vinberg. MR**1254931****[Wol84]**Joseph A. Wolf,*Spaces of constant curvature*, 5th ed., Publish or Perish, Inc., Houston, TX, 1984. MR**928600**

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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:
http://dx.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.