Local rigidity of 3-dimensional cone-manifolds

We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature $κ\in \{-1,0,1\}$ and cone-angles $\leq π$. 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 $L^2$-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.

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