Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery
type: Item
instance of: scholarly article
title: Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery
zbMATH Open document ID: 0919.57009
author: Craig Hodgson
author: Steven P. Kerckhoff
published in: Journal of Differential Geometry
publication date: 1999-08-31T00:00:00Z
review text: The authors extend the rigidity theorem of Weil and Garland for complete, finite volume hyperbolic manifolds of dimension at least 3 to a class of finite volume, orientable 3-dimensional hyperbolic cone-manifolds, that is, hyperbolic structures on 3-manifolds with cone-like singularities along a knot or link. Specifically, they show that such structures are locally rigid if the cone angles are fixed and all cone angles are at most \(2\pi\). Furthermore, they show that in this case, the deformation space of cone-manifold structures is locally parametrized by the set of cone angles. The main theorem proved by the authors is an \(L^2\) rigidity theorem (Theorem 1.1 of the paper) as follows: Theorem 1.1: Let \(\overline M\) be a finite volume, 3-dimensional hyperbolic cone-manifold, whose singular locus \(\Sigma\) is a knot or link. Let \(M\) denote the open, incomplete hyperbolic manifold \(\overline M-\Sigma\). If all the cone angles along \(\Sigma\) are at most \(2\pi\), then every closed \(L^2\) form in \(\Omega^1 (M,E)\) represents the trivial cohomology class in \(H^1(M;E)\). An infinitesimal deformation of a hyperbolic structure on the manifold \(M\) can be represented by an \(E\)-valued 1-form where \(E\) is the flat vector bundle of local Killing vector fields. The authors show that any infinitesimal deformation preserving cone angles can be represented by an \(L^2\) form. It then follows from theorem 1.1 that there are no infinitesimal deformations of the hyperbolic structure on \(M\) keeping the cone angles fixed. The proof of theorem 1.1 in the paper involves the use of Hodge theory and a Weitzenböck type formula. It is similar to the approach of Calabi, Weil and Matsushima-Murakami where it was shown that for a closed oriented manifold \(M\) every class in \(H^1(M;E)\) is represented by a unique harmonic form \(\omega\) which is both closed and co-closed and with the boundary term for the expansion of \((d\omega,d\omega) + (\delta\omega, \delta\omega)\) equ…
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57R65
zbMATH DE Number: 1187443
zbMATH Keywords: \(L^2\) rigidity theorem
zbMATH Keywords: hyperbolic structure
reviewed by: Ser Peow Tan
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214460606
OpenAlex ID: W1545838390
Wikidata QID: Q115174902
DOI: 10.4310/JDG/1214460606
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description: scientific article; zbMATH DE number 1187443