Pinching rigidity theorems for normal scalar curvature

TL;DR

This paper establishes pinching theorems for minimal submanifolds in spheres, showing that under certain curvature bounds, the normal bundle must be flat and classifying such submanifolds.

Contribution

It introduces new curvature pinching conditions involving the largest eigenvalue of shape operators and normal scalar curvature, leading to rigidity and classification results.

Findings

Normal bundle of the submanifold is flat under the pinching condition.

Provides classification of minimal submanifolds satisfying the curvature bounds.

Establishes a relation between eigenvalues of shape operators and normal scalar curvature.

Abstract

Let $M^n$ be an $n$-dimensional closed minimal submanifold immersed in the unit sphere $\mathbb{S}^{n+m}$. Denote by $S$ and $\rho^{\perp}$ the squared norm of the second fundamental form and the normal scalar curvature of $M^n$, respectively. Let $\{A^{\alpha}\}_{\alpha=n+1}^{n+m}$ be the shape operators of $M^n$ with respect to a local orthonormal normal frame. Denote by $\lambda_{1}$ the largest eigenvalue of the positive semi-definite symmetric matrix $\mathcal{A}=(\langle A^{\alpha},A^{\beta}\rangle)_{m\times m}$. We show that if $\lambda_{1}\leqslant n$ and $\rho^{\perp}\leqslant \left[{\sqrt{2}n(n-1)}\right]^{-1} \mathop{\inf}\limits_{p\in M}(n-\lambda_{1})(p)$, then $\rho^{\perp}\equiv 0$, which means the normal bundle of $M^n$ is flat, and further we give the classification of $M^n$.