Genus one minimal k-noids and saddle towers in H2×R

Abstract

For each k≥ 3, we construct a 1-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space H2×R with genus 1 and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus 1 and 2k ends in the quotient of H2×R by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature -4kπ. Finally, we also provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus 1 in quotients of H2×R by the action of a hyperbolic or parabolic translation.