Volume maximization and the extended hyperbolic space

Feng Luo, Jean Marc Schlenker

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1053-1068
Number of pages16
JournalProceedings of the American Mathematical Society
Volume140
Issue number3
DOIs
StatePublished - Jan 1 2012

Fingerprint

Hyperbolic Structure
Hyperbolic Space
Hyperbolic Metric
Decomposition
Spacelike Surface
De Sitter Space
Connected Sum
Totally Geodesic
Geometric Structure
Natural Extension
Projective plane
Critical point
Euclidean
Assignment
Angle
Decompose

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Proceedings of the American Mathematical Society, Vol. 140, No. 3, 01.01.2012, p. 1053-1068.

Research output: Contribution to journalArticle

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