Analytic saddle spheres in ${\mathbb{S}}^3$ are equatorial

Abstract

A theorem by Almgren establishes that any minimal $2$-sphere immersed in ${\mathbb{S}}^3$ 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 ${\mathbb{S}}^3$ that is saddle, i.e., of non-positive extrinsic curvature, must be an equator of ${\mathbb{S}}^3$. We remark that, contrary to Almgren's theorem, no geometric PDE is imposed on the surface. The result is not true for $C^{\infty }$ spheres.

17 pages, 9 figures