We prove the existence of complete minimal surfaces, in Euclidean 3-space, of arbitrary positive genus and least total absolute curvature that have precisely two ends: a catenoidal end and an Enneper-type end. This talk is based on a joint work with Rivu Bardhan, Indranil Biswas, and Pradip Kumar. https://arxiv.org/abs/2509.03925

We construct complete minimal surfaces with two ends in Euclidean 3-space. In particular, we prove the existence of such surfaces with arbitrary even number of genus which minimize the degree of the Gauss map for their genus. We introduce some remaining problems as well.