First stability eigenvalue characterization of Clifford hypersurfaces

ABSTRACT : The stability operator of a compact oriented minimal hypersurface Mn−1 ⊂ S is given by J = −∆ − ‖A‖ − (n − 1), where ‖A‖ is the norm of the second fundamental form. Let λ1 be the first eigenvalue of J and define β = −λ1 − 2(n − 1). In [S] Simons proved that β ≥ 0 for any non-equatorial minimal hypersurface M ⊂ S. In this paper we will show that β = 0 only for Clifford hypersurfaces. For minimal surfaces in S, let |M | denote the area ofM and let g denote the genus ofM . We will prove that β|M | ≥ 8π(g − 1). Moreover, if M is embedded, then we will prove that β ≥ g−1 g+1 . If in addition of the embeddeness condition we have that β < 1 then we will prove that |M | ≤ 16π 1−β . §