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

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.


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