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Laplacian eigenvalues and distances between subsets of a manifold. (English) Zbl 1032.58015

In the paper under review, the authors prove the following theorem: Let \(L\) be an analytic Laplacian on a metric measure space \((M,\mu)\), and let \(\lambda_{1}\) be the lower bound of the spectrum of \(L\) acting on the functions orthogonal to the constants. Then for \(X,Y\) disjoint Borel subsets of \(M\), we have: \[ \lambda_{1}\leq \frac{1}{\text{dist}^{2}(X,Y)}\Big(\cos h^{-1} (\sqrt{\frac{\mu(X^{c})\mu(Y^{c})} {\mu(X)\mu(Y)}})\Big)^{2} \] where \(X^{c}\) (resp. \(Y^{c}\)) is the complement of \(X\) (resp. \(Y\)). The authors introduce a new method to convert results on graph theoretic Laplacians into results concerning Laplacians in Analysis. They also prove several variants of their main theorem and give various examples. In particular, they illustrate their method by the example where \(M\) is a connected compact Riemannian manifold.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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