The spectrum of compact hypersurface in sphere

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\(\overline M \).