Minimal surfaces in a unit sphere pinched by intrinsic curvature and normal curvature

TL;DR

This paper introduces a special orthonormal frame for minimal surfaces in a sphere, revealing a key curvature relation and establishing pinching conditions on intrinsic and normal curvatures.

Contribution

It constructs a novel orthonormal frame simplifying shape operators and derives a fundamental curvature relation for minimal surfaces in spheres.

Findings

Established the relation K + K^N = 1 for positive Gauss curvature.

Derived pinching conditions on intrinsic and normal curvatures.

Provided a new framework for analyzing minimal surfaces in spheres.

Abstract

We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere $S^{n}$, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property $K+K^{N}=1$ for the Gauss curvature $K$ and the normal curvature $K^{N}$ if the Gauss curvature is positive. Moreover, using this property we obtain the pinching on the intrinsic curvature and normal curvature, the pinching on the normal curvature, respectively.