Rigidez de hipersuperfícies mínimas em Sn com curvatura de Ricci Constante

Abstract

Let M be a compact oriented minimal hypersurface of the unit n-dimensional sphere S . n In this paper we will point out that if the Ricci curvature of M is constant, then, we have n−3 n−2 that either Ric ≡−1 and M is isometric to an equator or, n is odd, Ric ≡−and M is √−√− isometric to S . Next, we will prove that there exists a positive number (n) 2 2 ×−S 2 2 such that if the Ricci curvature of a minimal hypersurface immersed by the rst eigenfunc- n−3 n−2 n−3 n−2 tions M satis es that −− (n) ≤−Ric ≤−−− (n) and the average of the scalar curvature n−3 n−2 is , then, the Ricci curvature of M must be constant and therefore M must be isometric √−√− .