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

Abstract

We establish a nice orthonormal frame field on a closed surfaceminimally immersed in a unit sphere $\mathbb{S}^{n}$, under which the shapeoperators take very simple forms. Using this frame field, we obtainan interesting property $K+K^{N}=1$ for the Gauss curvature $K$ andthe normal curvature $K^{N}$ if the Gauss curvature is positive.Moreover, using this property we obtain the pinching on theintrinsic curvature and normal curvature, the pinching on the normalcurvature, respectively.