INDEX AND TOTAL CURVATURE OF SURFACES WITH CONSTANT MEAN-CURVATURE

We prove an analogue, for surfaces with constant mean curvature in hyperbolic space, of a theorem of Fischer-Colbrie and Gulliver about minimal surfaces in Euclidean space. That is, for a complete surface M2 in hyperbolic 3-space with constant mean curvature 1, the (Morse) index of the operator L = delta - 2K is finite if and only if the total Gaussian curvature is finite.