First stability eigenvalue characterization of Clifford hypersurfaces

The stability operator of a compact oriented minimal hypersurface Mn-1 subset of S-n is given by J = -Delta - \\A\\(2) (n - 1), where A is the norm of the second fundamental form. Let lambda(1) be the first eigenvalue of J and de ne beta = lambda(1) - 2(n - 1). In 1968 Simons proved that 0 for any non-equatorial minimal hypersurface M S n. In this paper we will show that beta= 0 only for Clifford hypersurfaces. For minimal surfaces in S-3, let \M\ denote the area of M and let g denote the genus of M. We will prove that beta\M\ greater than or equal to 8pi ( g-1). Moreover, if M is embedded, then we will prove that beta greater than or equal to g-1/g+1. If in addition to the embeddeness condition we have that beta < 1, then we will prove that \M\ ≤ 16π/1-β.