The rigidity of Clifford torus $$S^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)$$

Abstract

In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS n+1 (1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n 2(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.