Minimal Surfaces in R with Dihedral Symmetry

We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most 2n+1 ends, and with symmetry group the natural Z2 extension of the dihedral group Dn. 1 2 The surfaces are constructed by proving existence of the conjugate surfaces. We extend this method to cases where the conjugate surface of the fundamental piece is noncompact and is not a graph over a convex plane domain.