Journal of Differential Geometry

Local rigidity of 3-dimensional cone-manifolds

Hartmut Weiss

Abstract

We study the local deformation space of 3-dimensional cone-manifold structures of constant curvature κ is an element of {.1, 0, 1} and cone-angles ≤ π. 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 L2-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 L2-cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms.

Article information

Source
J. Differential Geom., Volume 71, Number 3 (2005), 437-506.

Dates
First available in Project Euclid: 28 March 2006

https://projecteuclid.org/euclid.jdg/1143571990

Digital Object Identifier
doi:10.4310/jdg/1143571990

Mathematical Reviews number (MathSciNet)
MR2198808

Zentralblatt MATH identifier
1098.53038

Citation

Weiss, Hartmut. Local rigidity of 3-dimensional cone-manifolds. J. Differential Geom. 71 (2005), no. 3, 437--506. doi:10.4310/jdg/1143571990. https://projecteuclid.org/euclid.jdg/1143571990