The deformation theory of hyperbolic cone–$3$–manifolds with cone-angles less than $2\pi$

Abstract

We develop the deformation theory of hyperbolic cone–3–manifolds with cone-angles less than $2\pi$, that is, contained in the interval $\left(0,2\pi \right)$. In the present paper we focus on deformations keeping the topological type of the cone-manifold fixed. We prove local rigidity for such structures. This gives a positive answer to a question of A Casson.