Analytic saddle spheres in S3 are equatorial

Abstract

A theorem by Almgren establishes that any minimal 2-sphere immersed in is a totally geodesic equator. In this paper we give a purely geometric extension of Almgren’s result, by showing that any immersed, real analytic 2-sphere in that is saddle, i.e., of non-positive extrinsic curvature, must be an equator of . We remark that, contrary to Almgren’s theorem, no geometric PDE is imposed on the surface. The result is not true for spheres.