论文标题
在球体上的第二个拉普拉斯特征值的最大化
Maximization of the second Laplacian Eigenvalue on the Sphere
论文作者
论文摘要
对于$ s^m $的Laplacian的第二个非零特征值，我们证明了第二个非零特征值。对于$ s^{2} $，第二个非零特征值随着表面退化为两个不相交球而变得最大，这是由于Nadirashvili的结果，后来Petrides后来给出了另一个证明。对于较高的维球，类似的上限是由Girouard，Nadirashvili和Polterovich猜想的。我们确认猜想基于质量质量中心的工作和最新发展的方法，并为$ s^2 $提供了更简单的证明。
We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on $S^m$. For $S^{2}$, the second nonzero eigenvalue becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. For higher dimensional spheres, the analogous upper bound was conjectured by Girouard, Nadirashvili and Polterovich. Our method to confirm the conjecture builds on Petrides' work and recent developments on the hyperbolic center of mass and provides also a simpler proof for $S^2$.
