Rigidity of minimal hypersurfaces of spheres with constant ricci curvature

ABSTRACT: Let M be a compact oriented minimal hypersurface of the unit n-dimensional sphere S. In this paper we will point out that if the Ricci curvature of M is constant, then, we have that either Ric ≡ 1 andM is isometric to an equator or, n is odd,Ric ≡ n−3 n−2 andM is isometric to S n−1 2 ( √ 2 2 )×S n−1 2 ( √ 2 2 ). Next, we will prove that there exists a positive number ̄(n) such that if the Ricci curvature of a minimal hypersurface immersed by first eigenfunctions M satisfies that n−3 n−2 − ̄(n) ≤ Ric ≤ n−3 n−2 + ̄(n) and the average of the scalar curvature is n−3 n−2 , then, the ricci curvature of M must be constant and therefore M must be isometric to S n−1 2 ( √ 2 2 )× S n−1 2 ( √ 2 2 ). We will show that the condition on the first eigenvalue of the laplacian is necessary: for every ̄ > 0 there exist infinitely many minimal tori in S3 with |Ric| < ̄.