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Laplacian Eigenvalues and Distances Between Subsets of a Manifold
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Abstract:
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 $\epsilon>0$.
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