Abstract
In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|2 ≥ n. The natural generalization for this pinching is −(r + 2)S r+2 ≥ (n − r)S r > 0. Under this condition we shall extend such result for closed oriented hypersurface Σn of the Euclidean unit sphere \({\mathbb{S}^{n+1}}\) with null S r+1 mean curvature by showing that the index of r-stability, \({Ind_{\Sigma^n}^{r}}\), also satisfies \({Ind_{\Sigma^n}^{r}\ge n+3}\). Instead of the previous hypothesis if we consider \({\frac{S_{r+2}}{{S_r}}}\) constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces \({\Sigma^{n}\subset \mathbb{S}^{n+1}}\) with S r+1 = 0 and S r+2 < 0 have index of r-stability greater than or equal to 2n + 5.
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Communicated by D.V. Alekseevsky.
A. Barros was partially supported by CNPq, Brazil.
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Barros, A., Sousa, P. Estimate for index of hypersurfaces in spheres with null higher order mean curvature. Monatsh Math 160, 227–241 (2010). https://doi.org/10.1007/s00605-009-0123-1
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DOI: https://doi.org/10.1007/s00605-009-0123-1