An overview of curvature bounds and spectral theory of planar tessellations

Matthias Keller

doi:10.5802/acirm.55

Matthias Keller ¹

¹ Mathematisches Institut Friedrich Schiller Universität Jena 07743 Jena, Germany

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

Abstract

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.

Published online: 2014-11-12

Zbl: 06938603

DOI: 10.5802/acirm.55

Author's affiliations:

Matthias Keller ¹

¹ Mathematisches Institut Friedrich Schiller Universität Jena 07743 Jena, Germany

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

References
Cited by

[1] Michael Aizenman; Simone Warzel The canopy graph and level statistics for random operators on trees, Math. Phys. Anal. Geom., Volume 9 (2006) no. 4, p. 291-333 (2007) | DOI | MR | Zbl

[2] Frank Bauer; Bobo Hua; Matthias Keller On the $l^{p}$ spectrum of Laplacians on graphs, Adv. Math., Volume 248 (2013), pp. 717-735 | DOI | Zbl

[3] O. Baues; N. Peyerimhoff Curvature and geometry of tessellating plane graphs, Discrete Comput. Geom., Volume 25 (2001) no. 1, pp. 141-159 | DOI | MR | Zbl

[4] Oliver Baues; Norbert Peyerimhoff Geodesics in non-positively curved plane tessellations, Adv. Geom., Volume 6 (2006) no. 2, pp. 243-263 | DOI | MR | Zbl

[5] Ethan D. Bloch A characterization of the angle defect and the Euler characteristic in dimension 2, Discrete Comput. Geom., Volume 43 (2010) no. 1, pp. 100-120 | DOI | MR | Zbl

[6] Michel Bonnefont; Sylvain Golénia; Matthias Keller Eigenvalue asymptotics for Schrödinger operators on sparse graphs, preprint (2013) | Zbl

[7] Michel Bonnefont; Sylvain Golénia; Matthias Keller Eigenvalue asymptotics and unique continuation of eigenfunctions on planar graphs, preprint (2014)

[8] Jonathan Breuer; Matthias Keller Spectral analysis of certain spherically homogeneous graphs, Oper. Matrices, Volume 7 (2013) no. 4, pp. 825-847 | DOI | MR | Zbl

[9] Beifang Chen The Gauss-Bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature, Proc. Amer. Math. Soc., Volume 137 (2009) no. 5, pp. 1601-1611 | DOI | MR | Zbl

[10] Beifang Chen; Guantao Chen Gauss-Bonnet formula, finiteness condition, and characterizations of graphs embedded in surfaces, Graphs Combin., Volume 24 (2008) no. 3, pp. 159-183 | DOI | MR | Zbl

[11] Matt DeVos; Bojan Mohar An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture, Trans. Amer. Math. Soc., Volume 359 (2007) no. 7, p. 3287-3300 (electronic) | DOI | MR | Zbl

[12] Jozef Dodziuk Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc., Volume 284 (1984) no. 2, pp. 787-794 | DOI | MR | Zbl

[13] Jozef Dodziuk; Leon Karp Spectral and function theory for combinatorial Laplacians, Geometry of random motion (Ithaca, N.Y., 1987) (Contemp. Math.), Volume 73, Amer. Math. Soc., Providence, RI, 1988, pp. 25-40 | DOI | MR | Zbl

[14] Józef Dodziuk; Peter Linnell; Varghese Mathai; Thomas Schick; Stuart Yates Approximating $L^{2}$ -invariants and the Atiyah conjecture, Comm. Pure Appl. Math., Volume 56 (2003) no. 7, pp. 839-873 (Dedicated to the memory of Jürgen K. Moser) | DOI | MR | Zbl

[15] Harold Donnelly; Peter Li Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J., Volume 46 (1979) no. 3, pp. 497-503 http://projecteuclid.org/euclid.dmj/1077313570 | MR | Zbl

[16] Pasquale Joseph Federico Descartes on polyhedra, Sources in the History of Mathematics and Physical Sciences, 4, Springer-Verlag, New York-Berlin, 1982, viii+145 pages (A study of the ıt De solidorum elementis) | MR

[17] Koji Fujiwara Growth and the spectrum of the Laplacian of an infinite graph, Tohoku Math. J. (2), Volume 48 (1996) no. 2, pp. 293-302 | DOI | MR | Zbl

[18] Koji Fujiwara The Laplacian on rapidly branching trees, Duke Math. J., Volume 83 (1996) no. 1, pp. 191-202 | DOI | MR | Zbl

[19] Sylvain Golénia Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians, J. Funct. Anal., Volume 266 (2014) no. 5, pp. 2662-2688 | DOI | MR | Zbl

[20] M. Gromov Hyperbolic groups, Essays in group theory (Math. Sci. Res. Inst. Publ.), Volume 8, Springer, New York, 1987, pp. 75-263 | DOI | MR | Zbl

[21] Rainer Hempel; Jürgen Voigt The spectrum of a Schrödinger operator in $L_{p} (R^{ν})$ is $p$ -independent, Comm. Math. Phys., Volume 104 (1986) no. 2, pp. 243-250 http://projecteuclid.org/euclid.cmp/1104115001 | DOI | Zbl

[22] Yusuke Higuchi Combinatorial curvature for planar graphs, J. Graph Theory, Volume 38 (2001) no. 4, pp. 220-229 | DOI | MR | Zbl

[23] Bobo Hua; Jürgen Jost Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature II, to appear in Trans. Amer. Math. Soc. | Zbl

[24] Bobo Hua; Jürgen Jost; Shiping Liu Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature, to appear in J. Reine Angew. Math. | Zbl

[25] M. Ishida Pseudo-curvature of a graph, lecture at ’Workshop on topological graph theory’, Yokohama National University (1990)

[26] Matthias Keller The essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann., Volume 346 (2010) no. 1, pp. 51-66 | DOI | MR | Zbl

[27] Matthias Keller Curvature, geometry and spectral properties of planar graphs, Discrete Comput. Geom., Volume 46 (2011) no. 3, pp. 500-525 | DOI | MR | Zbl

[28] Matthias Keller; Daniel Lenz Agmon type estimates and purely discrete spectrum for graphs, preprint

[29] Matthias Keller; Norbert Peyerimhoff Cheeger constants, growth and spectrum of locally tessellating planar graphs, Math. Z., Volume 268 (2011) no. 3-4, pp. 871-886 | DOI | MR | Zbl

[30] Steffen Klassert; Daniel Lenz; Norbert Peyerimhoff; Peter Stollmann Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature, Proc. Amer. Math. Soc., Volume 134 (2006) no. 5, pp. 1549-1559 | DOI | MR | Zbl

[31] Steffen Klassert; Daniel Lenz; Peter Stollmann Discontinuities of the integrated density of states for random operators on Delone sets, Comm. Math. Phys., Volume 241 (2003) no. 2-3, pp. 235-243 | DOI | MR | Zbl

[32] H. P. McKean An upper bound to the spectrum of $Δ$ on a manifold of negative curvature, J. Differential Geometry, Volume 4 (1970), pp. 359-366 | DOI | MR | Zbl

[33] Bojan Mohar Many large eigenvalues in sparse graphs, European J. Combin., Volume 34 (2013) no. 7, pp. 1125-1129 | DOI | MR | Zbl

[34] Byung-Geun Oh Duality Properties of Strong Isoperimetric Inequalities on a Planar Graph and Combinatorial Curvatures, Discrete Comput. Geom., Volume 51 (2014) no. 4, pp. 859-884 | DOI | MR | Zbl

[35] Yoshiki Ohno; Hajime Urakawa On the first eigenvalue of the combinatorial Laplacian for a graph, Interdiscip. Inform. Sci., Volume 1 (1994) no. 1, pp. 33-46 | DOI | MR | Zbl

[36] Barry Simon Brownian motion, $L^{p}$ properties of Schrödinger operators and the localization of binding, J. Funct. Anal., Volume 35 (1980) no. 2, pp. 215-229 | DOI | Zbl

[37] David A. Stone A combinatorial analogue of a theorem of Myers, Illinois J. Math., Volume 20 (1976) no. 1, pp. 12-21 | MR | Zbl

[38] Karl-Theodor Sturm On the $L^{p}$ -spectrum of uniformly elliptic operators on Riemannian manifolds, J. Funct. Anal., Volume 118 (1993) no. 2, pp. 442-453 | DOI | MR | Zbl

[39] Liang Sun; Xingxing Yu Positively curved cubic plane graphs are finite, J. Graph Theory, Volume 47 (2004) no. 4, pp. 241-274 | DOI | MR | Zbl

[40] Wolfgang Woess A note on tilings and strong isoperimetric inequality, Math. Proc. Cambridge Philos. Soc., Volume 124 (1998) no. 3, pp. 385-393 | DOI | MR | Zbl

[41] Radoslaw Krzysztof Wojciechowski Stochastic completeness of graphs, ProQuest LLC, Ann Arbor, MI, 2008, 87 pages http://search.proquest.com/docview/304670648 Thesis (Ph.D.)–City University of New York | MR

[42] Lili Zhang A result on combinatorial curvature for embedded graphs on a surface, Discrete Math., Volume 308 (2008) no. 24, pp. 6588-6595 | DOI | MR | Zbl

[43] Andrzej Żuk On the norms of the random walks on planar graphs, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 5, pp. 1463-1490 | Numdam | MR | Zbl

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