Deformations of hyperbolic structures without the completeness condition (Q1067679)

This paper deals with incomplete deformations \(f\in Def_*M\) of a complete non-compact hyperbolic manifold M, vol M\(<\infty\), i.e. with homeomorphisms \(f: M\to M'\) onto hyperbolic manifolds M', vol M'\(<\infty\), which is incomplete but any geodesic going to an end of M' raises up to the maximal geodesic in the hyperbolic space \(H^ n\) \((f_ 1\approx f_ 2\) iff \(f_ 2f_ 1^{-1}\) is locally isometry). In dim M\(=3\) it is proved that the complex dimension of \(Def_*M\) equals the number of isolated ends of M. But in this case the subspace in \(Def_*M\) consisting of quasiconformal deformations consists (just as in the complete case) of one point. In dim \(M\geq 4\) it is proved that \(Def_*M\) consists of one point, i.e. there is strong rigidity of deformations, similar to the complete case. For a somewhat different approach to these problems (closely related with tiling problems of the hyperbolic space) see the author's paper ''Filling a space by polyhedra and deformations of incomplete hyperbolic structures'' (Russian), Sib. Mat. Zh. 27 (1986), English transl. in Sib. Math. J. 27 (1986) (to appear).