Abstract
Let M be a compact minimal hypersurface of sphere Sn+1(1). Let\(\overline M \) be H(r)-torus of sphere Sn+1(1). Assume they have the same constant mean curvature H, the result in [1] is that if\(Spec^0 (M,g) = Spec^0 (\overline M ,g)\), then for\(3 \leqslant n \leqslant 6, r^2 \leqslant \frac{{n - 1}}{n}\) or\(n \geqslant 6,r^2 \geqslant \frac{{n - 1}}{n}\), then M is isometric to\(\overline M \). We improved the result and prove that: if\(Spec^0 (M,g) = Spec^0 (\overline M ,g)\), then M is isometric to\(\overline M \). Generally, if\(Spec^p (M,g) = Spec^p (\overline M ,g)\), here p is fixed and satisfies that n(n−1)≠6p(n−p), then M is isometric to\(\overline M \).
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Supported by National Natural Science Foundation of China (10371047)
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Senlin, X., Qintao, D. & Dongmei, C. The spectrum of compact hypersurface in sphere. Anal. Theory Appl. 20, 288–293 (2004). https://doi.org/10.1007/BF02835296
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DOI: https://doi.org/10.1007/BF02835296