A family of higher genus complete minimal surfaces that includes the Costa-Hoffman-Meeks one

Abstract

In this paper, we construct a one-parameter family of minimal surfaces in the Euclidean $3$-space of arbitrarily high genus and with three ends. Each member of this family is immersed, complete and with finite total curvature. Another interesting property is that the symmetry group of the genus $k$ surfaces $Σ_{k,x}$ is the dihedral group with $4(k+1)$