Laplacian Eigenvalues and Distances Between Subsets of a Manifold
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In this paper we give a new method to convert results dealing with graph theoretic (or Markov chain) Laplacians into results concerning Laplacians in analysis, such as on Riemannian manifolds. We illustrate this method by using the results of \cite{chung_grigoryan_yau_2} to prove $$\lambda_1\le \frac{1}{{\rm dist}^2(X,Y)} \left(\cosh^{-1}\sqrt{\frac{\mu\mycomplement X \,\mu\mycomplement Y}{\mu X\,\mu Y} } \right)^2.$$ for % the (is the'' really necessary here?) $\lambda_1$ the first positive Neumann eigenvalue on a connected compact Riemannian manifold, and $X,Y$ any two disjoint sets (and where $\mycomplement X$ is the complement of $X$). This inequality has a version for the $k$-th positive eigenvalue (involving $k+1$ disjoint sets), and holds more generally for all analytic'' Laplacians described in \cite{chung_grigoryan_yau_2}. We show that this inequality is optimal to first order,'' in that it is impossible to obtain an inequality of this form with the right-hand-side divided by $1+\epsilon$ for any fixed constant $\epsilon>0$.