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The spectrum of compact hypersurface in sphere

Authors:	Email author" target="_blank">Xu?Senlin Email author Deng?Qintao Chen?Dongmei

Institution:	(1) Department of Mathematics, Central China Normal University, 430079 Wuhan, Hubei, P. R. China;(2) Department of Mathematics, University of Science and Technology, 230026 Heifei, P. R. China

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 $\overline M$ . Supported by National Natural Science Foundation of China (10371047)

Keywords:	Laplace operator spectrum isometric SPHERE HYPERSURFACE COMPACT fixed improved isometric result constant mean curvature torus compact minimal hypersurface sphere
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