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 $\Sigma_{k,x}$ is the dihedral group with $4(k+1)$ elements. Moreover, in particular, for $|x|=1$ we find the family of the Costa-Hoffman-Meeks embedded minimal surfaces, which have two catenoidal ends and a middle flat end.