MINIMAL-SURFACES WITH CATENOID ENDS

A substantive part of the recent activity in the held of minimal surfa ce theory has been the construction of new complete minimal surfaces i mmersed in R(3). One approach in constructing new examples is to incre ase the genus of known minimal surfaces. In this paper, we do precisel y this for certain minimal surfaces of finite total curvature whose en ds are asymptotic to catenoids. We prove existence of surfaces of posi tive genus based on those in genus zero, with the feature that these h igher genus examples maintain all the symmetry of their genus-zero cou nterparts. In these proofs we use the conjugate minimal surface constr uction and the maximum principle for minimal surfaces.