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$\overline M$.