arXiv 2007.067342020-07-13math.SPmath.DG

高维手术与Steklov特征值

Higher dimensional surgery and Steklov eigenvalues

Han Hong

备注：: 17 pages

摘要：

我们证明了对于维数至少为$3$且具有非空边界的紧致黎曼流形，我们可以通过对余维$2$或更高维的流形进行修改，同时保持Steklov谱几乎不变。这表明，当考虑尺寸$3$和更高的Steklov特征值的形状优化问题时，域的拓扑结构中的某些变化没有影响。我们的结果将[FS 2]中的一维手术推广到更高维的手术和更高的特征值。在[FS 2]中证明了单位球不极大化$\mathbb{R}^n$中可收缩域的第一个非零归一化Steklov特征值，对于$n \geq 3$ .我们表明，这也是正确的更高的Steklov特征值。使用类似的想法，我们表明，在$\mathbb{R}^n$，对于$n\geq 3$，$j$ -th归一化Steklov特征值不是最大化的限制由一系列的可收缩域退化到$j$单位球的不相交的工会，在相反的情况下，在尺寸$2$ [GP 1]。

英文摘要：

We show that for compact Riemannian manifolds of dimension at least $3$ with nonempty boundary, we can modify the manifold by performing surgeries of codimension $2$ or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions $3$ and higher. Our result generalizes the 1-dimensional surgery in [FS2] to higher dimensional surgeries and to higher eigenvalues. It is proved in [FS2] that the unit ball does not maximize the first nonzero normalized Steklov eigenvalue among contractible domains in $\mathbb{R}^n$, for $n \geq 3$. We show that this is also true for higher Steklov eigenvalues. Using similar ideas we show that in $\mathbb{R}^n$, for $n\geq 3$, the $j$-th normalized Steklov eigenvalue is not maximized in the limit by a sequence of contractible domains degenerating to the disjoint union of $j$ unit balls, in contrast to the case in dimension $2$ [GP1].

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