ON THE INDEX OF CONSTANT MEAN-CURVATURE-1 SURFACES IN HYPERBOLIC SPACE

We show that the index of a constant mean curvature I surface in hyper bolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass repr esentation. We give three applications of this observation. Firstly, i t allows us to explicitly compute the index of the catenoid cousins an d some other examples. Secondly, it allows us to be able to apply a me thod similar to that of Choe (using Killing vector fields on minimal s urfaces in Euclidean 3-space) to our case as well, resulting in lower bounds of index for other examples. And thirdly, it allows us to give a more direct proof of the result by do Carmo and Silveira that if a c onstant mean curvature 1 surface in hyperbolic 3-space has finite tota l curvature, then it has finite index. Finally, we show that for any c onstant mean curvature 1 surfaces in hyperbolic 3-space that have been constructed via a correspondence to minimal surfaces in Euclidean 3-s pace, we can take advantage of this correspondence to find a lower bou nd for its index.