Minimal Surfaces in a Sphere and the Ricci Condition

Abstract

In this paper we deal with minimal surfaces in a sphere which are locally isometric to a minimal surface in S3. We prove that a minimal surface in a sphere is locally isometric to a minimal surface in S3 if the curvature ellipse has constant and positive eccentricity. Moreover, we prove the following rigidity result: a compact minimal surface M in Sm, m ≤ 6, cannot be locally isometric to a minimal surface in S3 unless M already lies in S3 or M is flat and lies in S5.