The topology of large H-surfaces bounded by a convex curve

We shall consider embedded compact surfaces M of constant non-zero mean cur vature H (N-surfaces) in hyperbolic space H-3. Let L denote a horosphere of H-3. Assume that M is contained in the horoball bounded by L and that the boundary of M is a strictly convex Jordan curve Gamma in L. We establish th e following: (i) case H > 1. There is an h(Gamma), depending only on the geometry of Gam ma, such that whenever M is a H-surface bounded by Gamma, with 1 < H < h(Ga mma), then M is topologically a disk. (ii) case H less than or equal to 1. Then M is a graph over the domain Omeg a subset of L bounded by Gamma with respect to the geodesics orthogonal to Omega; in particular, M is topologically a disk. (C) 2000 Editions scientif iques et medicales Elsevier SAS.