Deforming Euclidean cone 3-manifolds

Given a closed orientable Euclidean cone 3-manifold C with cone angles ≤ π and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles < π. We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbolic ones and we also deduce global rigidity for Euclidean cone structures.