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The deformation theory of hyperbolic and Euclidean cone-manifolds with allcone angles less then 2{\pi} plays an important role in many problems in lowdimensional topology and in the geometrization of 3-manifolds. Furthermore,various old conjectures dating back to Stoker about the moduli of convexhyperbolic and Euclidean polyhedra can be reduced to the study of deformationsof cone-manifolds by doubling a polyhedron across its faces. This deformationtheory has been understood by Hodgson and Kerckhoff when the singular set hasno vertices, and by Wei{\ss} when the cone angles are less than {\pi}. We provehere an infinitesimal rigidity result valid for cone angles less than 2{\pi},stating that infinitesimal deformations which leave the dihedral angles fixedare trivial in the hyperbolic case, and reduce to some simple deformations inthe Euclidean case. The method is to treat this as a problem concerning thedeformation theory of singular Einstein metrics, and to apply analytic methodsabout elliptic operators on stratified spaces. This work is an importantingredient in the local deformation theory of cone-manifolds by the secondauthor, see also the concurrent work by Wei{\ss}.

Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra