Minimal surfaces of constant curvature in $S^ n$

Abstract:

In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in ${S^n}$ of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in ${S^n}$ of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in ${S^n}$ and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into ${S^n}$.