DEFORMATION OF THREE-DIMENSIONAL HYPERBOLIC CONE STRUCTURES: THE NONCOLLAPSING CASE

Author(s)

Barreto, Alexandre Paiva ^[1]

Affiliation

^[1] Univ Fed Sao Carlos, Dept Matemat, BR-13565905 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: PACIFIC JOURNAL OF MATHEMATICS; v. 268, n. 1, p. 1-21, MAR 2014.
• Web of Science Citations: 0

Abstract

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the length of the singularity remains uniformly bounded over the deformation. Let (M-i,M- p(i)) be a sequence of pointed hyperbolic cone manifolds with cone angles of at most 2 pi and topological type (M, Sigma), where M is a closed, orientable and irreducible 3-manifold and Sigma an embedded link in M. Assuming that the length of the singularity remains uniformly bounded, we prove that either the sequence M-i collapses and M is Seifert fibered or a Sol manifold, or the sequence M-i does not collapse and, in this case, a subsequence of (M-i,M- p(i)) converges to a complete three dimensional Alexandrov space endowed with a hyperbolic metric of finite volume on the complement of a finite union of quasigeodesics. We apply this result to a question proposed by Thurston and to provide universal constants for hyperbolic cone structures when Sigma is a small link in M. (AU)

• FAPESP's process: 09/16234-5 - Geometric Cone-Structures on Manifolds of dimension 2 and 3
• Grantee: Alexandre Paiva Barreto
• Support Opportunities: Scholarships in Brazil - Post-Doctoral