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October, 2000 Laplacian Eigenvalues and Distances Between Subsets of a Manifold
Joel Friedman, Jean-Pierre Tillich
J. Differential Geom. 56(2): 285-299 (October, 2000). DOI: 10.4310/jdg/1090347645

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 [6] to prove

$$ \λ 1\leq {1/over dist2(X,Y)} /Big(\cosh -1\sqrt {\mu Xc\mu Yc\over \mu X\mu Y}\Big)2. $$

for λ1 the first positive Neumann eigenvalue on a connected compact Riemannian manifold, and X, Y any two disjoint sets (and where Xc 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 [6]. 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 + for any fixed constant ∊ > 0.

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Joel Friedman. Jean-Pierre Tillich. "Laplacian Eigenvalues and Distances Between Subsets of a Manifold." J. Differential Geom. 56 (2) 285 - 299, October, 2000. https://doi.org/10.4310/jdg/1090347645

Information

Published: October, 2000
First available in Project Euclid: 20 July 2004

zbMATH: 1032.58015
MathSciNet: MR1863018
Digital Object Identifier: 10.4310/jdg/1090347645

Rights: Copyright © 2000 Lehigh University

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Vol.56 • No. 2 • October, 2000
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