A Characterization of Isoparametric Surfaces in Space Forms via Minimal Surfaces

Abstract

We give a characterization of isoparametric surfaces in three dimensional space forms with the help of minimal surfaces. Let us consider a surface in a three dimensional space form with the following property: through each point of the surface pass three curves such that the ruled orthogonal surface determined by each of them is minimal. We prove that necessarily the initial surface is isoparametric. It is also shown, that the curves are necessarily geodesics. On other hand, using the classification of isoparametric surfaces it is possible to prove that they have the above property along every geodesic. So, we have a characterization. As a preliminary result we prove that a surface with two geodesics, through every point, which are helices of the ambient, has parallel shape operator.