On rigidity of Clifford torus in a unit sphere

We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S ⁿ⁺¹ satisfying Sf ₄ − f ₃² ≤ $ \tfrac{1} {n} $ \tfrac{1} {n} S ³, where S is the squared norm of the second fundamental form of M, and f _k $ \sum\limits_i {\lambda _i^k } $ \sum\limits_i {\lambda _i^k } and λ _i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n+δ(n), then S ≡ n, i.e., M is one of the Clifford torus $ S^k (\sqrt {\tfrac{k} {n}} ) \times S^{n - k} (\sqrt {\tfrac{{n - k}} {n}} ) $ S^k (\sqrt {\tfrac{k} {n}} ) \times S^{n - k} (\sqrt {\tfrac{{n - k}} {n}} ) for 1 ≤ k ≤ n − 1. Moreover, we prove that if S is a constant, then there exists a positive constant τ (n) (≥ n − $ \tfrac{2} {3} $ \tfrac{2} {3} depending only on n such that if n ≤ S < n+ τ(n), then S ≡ n, i.e., M is a Clifford torus.