Minimal surfaces of constant curvature in sⁿ

In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in Sn 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 Sn of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in Sn and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into Sn. © 1985 American Mathematical Society.

Duke Scholars

Author Robert Bryant Mathematics