Minimal Surfaces in the Three-Dimensional Sphere and Minimal Hypersurfaces of Type Number Two

TL;DR

This paper characterizes minimal surfaces in the 3-sphere using principal parameters and classifies certain minimal hypersurfaces of type number two, linking them to minimal surfaces in Euclidean space or the sphere.

Contribution

It introduces canonical principal parameters for minimal surfaces in the 3-sphere and establishes a classification for bi-umbilical hypersurfaces of type number two.

Findings

Minimal surfaces in the 3-sphere are determined by their normal curvature function satisfying the Sinh-Poisson equation.

Bi-umbilical hypersurfaces of type number two are classified.

Minimal hypersurfaces of type number two with involutive distribution are generated by minimal surfaces in Euclidean space or the sphere.

Abstract

We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of type number two. We prove that any minimal hypersurface of type number two with involutive distribution is generated by a minimal surface in the three-dimensional Euclidean space, or in the three dimensional sphere. Thus we prove that the theory of minimal hypersurfaces of type number two with involutive distribution is locally equivalent to the theory of minimal surfaces in the three dimensional Euclidean space or in the three-dimensional sphere.