An overview of curvature bounds and spectral theory of planar tessellations

Keller, Matthias

doi:10.5802/acirm.55

Keller, Matthias ¹

¹ Mathematisches Institut Friedrich Schiller Universität Jena 07743 Jena, Germany

Actes des rencontres du CIRM, Tome 3 (2013) no. 1, pp. 61-68.

Résumé

We give a survey about the spectral consequences of upper bounds on the curvature on planar tessellating graphs. We first discuss spectral bounds and then put a particular focus on uniformly decreasing curvature. This case is characterized by purely discrete spectrum and we further present eigenvalue asymptotics and exponential decay of eigenfunctions. We then discuss absence of compactly supported eigenfunctions and dependence of the spectrum of the Laplacian on the underlying $ℓ^{p}$ space.

Publié le : 2014-11-13

Zbl

DOI : 10.5802/acirm.55

Affiliations des auteurs :

Keller, Matthias ¹

¹ Mathematisches Institut Friedrich Schiller Universität Jena 07743 Jena, Germany

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Keller, Matthias. An overview of curvature bounds and spectral theory of planar tessellations. Actes des rencontres du CIRM, Tome 3 (2013) no. 1, pp. 61-68. doi : 10.5802/acirm.55. http://www.numdam.org/articles/10.5802/acirm.55/

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