### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1053-1068 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 140 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*140*(3), 1053-1068. https://doi.org/10.1090/S0002-9939-2011-10941-9

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*Proceedings of the American Mathematical Society*, vol. 140, no. 3, pp. 1053-1068. https://doi.org/10.1090/S0002-9939-2011-10941-9

**Volume maximization and the extended hyperbolic space.** / Luo, Feng; Schlenker, Jean Marc.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Volume maximization and the extended hyperbolic space

AU - Luo, Feng

AU - Schlenker, Jean Marc

PY - 2012/1/1

Y1 - 2012/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=82255191463&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82255191463&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2011-10941-9

DO - 10.1090/S0002-9939-2011-10941-9

M3 - Article

AN - SCOPUS:82255191463

VL - 140

SP - 1053

EP - 1068

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 3

ER -