Volume maximization and the extended hyperbolic space

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space the natural extension of hyperbolic space by the de Sitter space except for the degenerate case where all simplices are Euclidean in a generalized sense. Those extended hyperbolic structures can realize geometrically a decomposition of the manifold as the connected sum of components admitting a complete hyperbolic metric, along embedded spheres (or projective planes) which are totally geodesic space-like surfaces in the de Sitter part of the extended hyperbolic structure.