Index bounds for minimal hypersurfaces of the sphere. (English) Zbl 1209.53052

Savo, Alessandro

Indiana Univ. Math. J. 59, No. 3, 823-838 (2010).

Summary: We consider a compact, orientable minimal hypersurfaces of the unit sphere and prove a comparison theorem between the spectrum of the stability operator and that of the Laplacian on 1 -forms. As a corollary, we show that the index is bounded below by a linear function of the first Betti number; in particular, if the first Betti number is large, then the immersion is highly unstable.

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58C40	Spectral theory; eigenvalue problems on manifolds

Keywords:

stability operator; Laplacian on 1-forms; Betti number