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arXiv 1010.0953math.DG
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常数量曲率超曲面的Jacobi算子的第二个eigenvalue

Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature

Haizhong Li, Xianfeng Wang

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摘要：

设$x:M\to\mathbb{S}^{n+1}(1)$是单位球面$\mathbb{S}^{n+1}(1),~n\geq 5$中具有常数量曲率$n(n-1)r,~r\geq 1$的n维紧致超曲面.我们知道，这样的超曲面可以被表征为平均曲率$H$的积分$\int_MH dv$的变分问题的临界点。本文给出了$M$的Jacobi算子$J_s$的第二特征值的最优上界。当$r>1$时，当且仅当$M$是全脐非全测地线时，当$r=1$时，当$M$是黎曼积$\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leq n-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$时，得到了上界.

英文摘要：

Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimal upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$. Moreover, when $r>1$, the bound is attained if and only if $M$ is totally umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is the Riemannian product $\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leq n-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.

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