Local rigidity of 3-dimensional cone-manifolds

We investigate the local deformation space of 3-dimensional conemanifold structures of constant curvature κ ∈ {−1, 0, 1} and coneangles≤ 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 Lcohomology 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-cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms.