First stability eigenvalue characterization of CliÞord hypersurfaces

ABSTRACT: The stability operator of a compact oriented minimal hypersurface Mn−1 ⊂ Sn is given by J = − ∆ − ‖A‖2 − (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 ⊂ Sn. In this paper we will show that β = 0 only for Clifford hypersurfaces. For minimal surfaces in S3, let |M | denote the area ofM and let g denote the genus ofM. We will prove that β|M | ≥ 8pi(g − 1). Moreover, if M is embedded, then we will prove that β ≥ g−1g+1. If in addition of the embeddeness condition we have that β < 1 then we will prove that |M | ≤ 16pi1−β. §1 Introduction & preliminaries In 1968, [S], James Simons proved an estimate for the first eigenvalue of the stability opera-tor on any minimal hypersurface Mn−1 ⊂ Sn. In this paper we will show that this estimat