Local rigidity of 3-dimensional cone-manifolds

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.