Local rigidity of 3-dimensional cone-manifolds

We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature \(\kappa \in \{-1,0,1\}\) and cone-angles \(\leq \pi\). Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main...

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Bibliographic Details
Published inarXiv.org
Main Author Weiss, Hartmut
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.04.2005
Subjects
Curvature
Deformation
Homology
Manifolds (mathematics)
Rigidity
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Summary:We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature \(\kappa \in \{-1,0,1\}\) and cone-angles \(\leq \pi\). Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem for the first \(L^2\)-cohomology group of the smooth part of the cone-manifold with coefficients in the flat bundle of infinitesimal isometries. We conclude local rigidity from this. In the Euclidean case we prove that the first \(L^2\)-cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms.
ISSN:2331-8422