First nonzero eigenvalue of a pseudo-umbilical hypersurface in the unit sphere |
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Authors: | Majid Ali Choudhary |
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Institution: | 1. Department of Mathematics, Jamia Millia Islamia, Jamia Nagar, New Delhi, 110025, India
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Abstract: | S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ 1 of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S n+1. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S n+1 with λ 1 = n is either isometric to the sphere S n or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate. |
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