论文标题
较高维度的手术和steklov特征值
Higher dimensional surgery and Steklov eigenvalues
论文作者
论文摘要
我们表明，对于尺寸紧凑的riemannian歧管至少$ 3 $具有非空边界，我们可以通过执行Codimension $ 2 $或更高的手术来修改歧管，同时保持steklov Spectrum几乎没有变化。这表明，在考虑对Steklov特征值的形状优化问题$ 3 $及更高时，域拓扑的某些变化没有效果。我们的结果将[FS2]中的一维手术概括为更高的维度手术和更高的特征值。在[FS2]中证明，单位球不会最大化$ \ Mathbb {r}^n $中的合同域中的第一个非零归一化steklov eigenvalue，以$ n \ geq 3 $。我们表明，对于更高的Steklov特征值也是如此。使用类似的想法，我们表明，在$ \ mathbb {r}^n $中，对于$ n \ geq 3 $，$ j $ th的标准化的steklov eigenvalue并未通过一系列可违约的域名为$ j $单位球的不相交联盟，与$ j $单位球的不相交，在disemension $ 2 $ 2 $ 2 $ [gp1 $ 2 $ 2 $ [gp1]中最大限度地限制。
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].
