A characterization of riemann’s minimal surfaces

Abstract

We prove that Riemann’s minimal surfaces are the only properly embedded minimal tori with two planar ends in ℝ3/T, where T is the group generated by a nontrivial translation in ℝ3. In the proof of this result we find all the properly immersed minimal tori with two parallel embedded planar ends. The space of such surfaces is described by regular curves, parameterized by ]R, in the moduli space of conformal structures on a topological torus. Except in the case of Riemann’s minimal surfaces, these curves contain points which yield minimal surfaces with vertical flux, and hence the surfaces are not embedded. © 1997 Journal of Differential Geometry. © 1997 Applied Probability Trust.