A substantive part of the recent
activity in the field of minimal surface theory has been the construction of new
complete minimal surfaces immersed in ℝ3. One approach in constructing new
examples is to increase the genus of known minimal surfaces. In this paper, we do
precisely this for certain minimal surfaces of finite total curvature whose ends are
asymptotic to catenoids. We prove existence of surfaces of positive genus based on
those in genus zero, with the feature that these higher genus examples maintain all
the symmetry of their genus-zero counterparts. In these proofs we use the
conjugate minimal surface construction and the maximum principle for minimal
surfaces.
