Geometric and Spectral Consequences of Curvature Bounds on Tessellations

Chapter
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 2184)

Abstract

This chapter focuses on geometric and spectral consequences of curvature bounds. Several of the results presented here have analogues in Riemannian geometry but in some cases one can go even beyond the Riemannian results and there also striking differences. The geometric setting of this chapter are tessellations and the curvature notion arises as a combinatorial quantity which can be interpreted as an angular defect and goes back to Descartes. First, we study the geometric consequences of curvature bounds. Here, a discrete Gauss–Bonnet theorem provides a starting point from which various directions shall be explored. These directions include analogues of a theorem of Myers, a Hadamard–Cartan theorem, volume growth bounds, strong isoperimetric inequalities and Gromov hyperbolicity. Secondly, we investigate spectral properties of the Laplacian which are often consequences of the geometric properties established before. For example we present analogues to a theorem of McKean about the spectral gap, a theorem by Donnelly-Li about discrete spectrum, we discuss the phenomena of compactly supported eigenfunctions and briefly elaborate on stability of the ℓ2 spectrum for the Laplacian on ℓ p .

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Notes

Acknowledgements

MK enjoyed the hospitality of C.I.R.M. and acknowledges the financial support of the German Science Foundation (DFG).

References

  1. 1.
    Aizenman, M., Warzel, S.: The canopy graph and level statistics for random operators on trees. Math. Phys. Anal. Geom. 9(4), 291–333 (2007) (2006). doi:10.1007/s11040-007-9018-3. http://dx.doi.org/10.1007/s11040-007-9018-3
  2. 2.
    Bartholdi, L., Ceccherini-Silberstein, T.G.: Salem numbers and growth series of some hyperbolic graphs. Geom. Dedicata 90, 107–114 (2002). http://dx.doi.org/10.1023/A:1014902918849 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bauer, F., Hua, B., Keller, M.: On the l p spectrum of Laplacians on graphs. Adv. Math. 248, 717–735 (2013). http://dx.doi.org/10.1016/j.aim.2013.05.029 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bauer, F., Keller, M., Wojciechowski, R.K.: Cheeger inequalities for unbounded graph Laplacians. J. Eur. Math. Soc. (JEMS) 17(2), 259–271 (2015). http://dx.doi.org/10.4171/JEMS/503
  5. 5.
    Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25(1), 141–159 (2001). http://dx.doi.org/10.1007/s004540010076 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243–263 (2006). http://dx.doi.org/10.1515/ADVGEOM.2006.014 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bonnefont, M., Golénia, S., Keller, M.: Eigenvalue asymptotics and unique continuation of eigenfunctions on planar graphs (2014). PreprintGoogle Scholar
  8. 8.
    Bonnefont, M., Golénia, S., Keller, M.: Eigenvalue asymptotics for Schrödinger operators on sparse graphs. Ann. Inst. Fourier (Grenoble) 65(5), 1969–1998 (2015). http://aif.cedram.org/item?id=AIF_2015__65_5_1969_0
  9. 9.
    Breuer, J., Keller, M.: Spectral analysis of certain spherically homogeneous graphs. Oper. Matrices 7(4), 825–847 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Byung-Geun, O.: On the number of vertices of positively curved planar graphs. Discret. Math. 340(6), 1300–1310 (2017). doi:10.1016/j.disc.2017.01.025, ISSN:0012-365X, http://dx.doi.org/10.1016/j.disc.2017.01.025
  11. 11.
    Cannon, J.W., Wagreich, P.: Growth functions of surface groups. Math. Ann. 293(2), 239–257 (1992). http://dx.doi.org/10.1007/BF01444714 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cao, J.: Cheeger isoperimetric constants of Gromov-hyperbolic spaces with quasi-poles. Commun. Contemp. Math. 2(4), 511–533 (2000). http://dx.doi.org/10.1142/S0219199700000232 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chen, B.: The Gauss-Bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature. Proc. Am. Math. Soc. 137(5), 1601–1611 (2009). http://dx.doi.org/10.1090/S0002-9939-08-09739-6 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Chen, B., Chen, G.: Gauss-Bonnet formula, finiteness condition, and characterizations of graphs embedded in surfaces. Graphs Combinatorics 24(3), 159–183 (2008). http://dx.doi.org/10.1007/s00373-008-0782-z MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    DeVos, M., Mohar, B.: An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture. Trans. Am. Math. Soc. 359(7), 3287–3300 (electronic) (2007). doi:http://dx.doi.org/10.1090/S0002-9947-07-04125-6
  16. 16.
    Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984). http://dx.doi.org/10.2307/1999107 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. In: From Local Times to Global Geometry, Control and Physics (Coventry, 1984/85). Pitman Research Notes in Mathematics Series, vol. 150, pp. 68–74. Longman Sci. Tech., Harlow (1986)Google Scholar
  18. 18.
    Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Geometry of Random Motion (Ithaca, N.Y., 1987). Contemporary Mathematics, vol. 73, pp. 25–40. American Mathematical Society, Providence (1988). http://dx.doi.org/10.1090/conm/073/954626
  19. 19.
    Dodziuk, J., Linnell, P., Mathai, V., Schick, T., Yates, S.: Approximating L 2-invariants and the Atiyah conjecture. Commun. Pure Appl. Math. 56(7), 839–873 (2003). http://dx.doi.org/10.1002/cpa.10076. Dedicated to the memory of Jürgen K. Moser
  20. 20.
    Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46(3), 497–503 (1979). http://projecteuclid.org/euclid.dmj/1077313570 MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Federico, P.J.: Descartes on polyhedra. Sources in the History of Mathematics and Physical Sciences, vol. 4. Springer, New York/Berlin (1982). A study of the ıt De solidorum elementisGoogle Scholar
  22. 22.
    Floyd, W.J., Plotnick, S.P.: Growth functions on Fuchsian groups and the Euler characteristic. Invent. Math. 88(1), 1–29 (1987). http://dx.doi.org/10.1007/BF01405088 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fujiwara, K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku Math. J. (2) 48(2), 293–302 (1996). http://dx.doi.org/10.2748/tmj/1178225382
  24. 24.
    Fujiwara, K.: The Laplacian on rapidly branching trees. Duke Math. J. 83(1), 191–202 (1996). http://dx.doi.org/10.1215/S0012-7094-96-08308-8 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gromov, M.: Hyperbolic groups. In: Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987). http://dx.doi.org/10.1007/978-1-4613-9586-7_3
  26. 26.
    Häggström, O., Jonasson, J., Lyons, R.: Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. 30(1), 443–473 (2002). http://dx.doi.org/10.1214/aop/1020107775 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hempel, R., Voigt, J.: The spectrum of a Schrödinger operator in L p(R ν) is p-independent. Commun. Math. Phys. 104(2), 243–250 (1986). URL http://projecteuclid.org/euclid.cmp/1104115001
  28. 28.
    Higuchi, Y.: Combinatorial curvature for planar graphs. J. Graph Theory 38(4), 220–229 (2001). http://dx.doi.org/10.1002/jgt.10004 MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Higuchi, Y., Shirai, T.: Isoperimetric constants of (d, f)-regular planar graphs. Interdiscip. Inform. Sci. 9(2), 221–228 (2003). http://dx.doi.org/10.4036/iis.2003.221
  30. 30.
    Hua, B., Jost, J., Liu, S.: Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature. J. Reine Angew. Math. 700, 1–36 (2015). http://dx.doi.org/10.1515/crelle-2013-0015 MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Ishida, M.: Pseudo-curvature of a graph. Lecture at ‘Workshop on topological graph theory’, Yokohama National University (1990)Google Scholar
  32. 32.
    Keller, M.: The essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), 51–66 (2010). http://dx.doi.org/10.1007/s00208-009-0384-y MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Keller, M.: Curvature, geometry and spectral properties of planar graphs. Discrete Comput. Geom. 46(3), 500–525 (2011). http://dx.doi.org/10.1007/s00454-011-9333-0 MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Keller, M.: Intrinsic metrics on graphs – A survey. In: Mugnolo, D. (ed.) Mathematical Technology of Networks (Proc. Bielefeld 2013). Proceedings in Mathematics and Statistics. Springer, New York (2014)Google Scholar
  35. 35.
    Keller, M., Peyerimhoff, N.: Cheeger constants, growth and spectrum of locally tessellating planar graphs. Math. Z. 268(3–4), 871–886 (2011). http://dx.doi.org/10.1007/s00209-010-0699-0 MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Keller, M., Peyerimhoff, N., Pogorzelski, F.: Sectional curvature of polygonal complexes with planar substructures. Adv. Math. 307, 1070–1107 (2017)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Keller, M., Lenz, D.: Agmon type estimates and purely discrete spectrum for graphs. PreprintGoogle Scholar
  38. 38.
    Keller, M., Liu, S., Peyerimhoff, N.: A note on eigenvalue bounds for non-compact manifolds, Preprint 2017, arXiv:1706.02437Google Scholar
  39. 39.
    Klassert, S., Lenz, D., Stollmann, P.: Discontinuities of the integrated density of states for random operators on Delone sets. Commun. Math. Phys. 241(2–3), 235–243 (2003)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Klassert, S., Lenz, D., Peyerimhoff, N., Stollmann, P.: Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature. Proc. Am. Math. Soc. 134(5), 1549–1559 (2006). http://dx.doi.org/10.1090/S0002-9939-05-08103-7 MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009). With a chapter by James G. Propp and David B. WilsonGoogle Scholar
  42. 42.
    McKean, H.P.: An upper bound to the spectrum of Δ on a manifold of negative curvature. J. Differ. Geom. 4, 359–366 (1970)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Mohar, B.: Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra Appl. 103, 119–131 (1988). http://dx.doi.org/10.1016/0024-3795(88)90224-8
  44. 44.
    Mohar, B.: Some relations between analytic and geometric properties of infinite graphs. Discrete Math. 95(1–3), 193–219 (1991). http://dx.doi.org/10.1016/0012-365X(91)90337-2. Directions in infinite graph theory and combinatorics (Cambridge, 1989)
  45. 45.
    Montenegro, R., Tetali, P.: Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1(3), x+121 (2006)Google Scholar
  46. 46.
    Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Nicholson, R., Sneddon, J.: New graphs with thinly spread positive combinatorial curvature. N. Z. J. Math. 41, 39–43 (2011)MathSciNetMATHGoogle Scholar
  48. 48.
    Oh, B.G.: Duality properties of strong isoperimetric inequalities on a planar graph and combinatorial curvatures. Discrete Comput. Geom. 51(4), 859–884 (2014). http://dx.doi.org/10.1007/s00454-014-9592-7 MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Ohno, Y., Urakawa, H.: On the first eigenvalue of the combinatorial Laplacian for a graph. Interdiscip. Inform. Sci. 1(1), 33–46 (1994). http://dx.doi.org/10.4036/iis.1994.33 MathSciNetMATHGoogle Scholar
  50. 50.
    Simon, B.: Brownian motion, L p properties of Schrödinger operators and the localization of binding. J. Funct. Anal. 35(2), 215–229 (1980). http://dx.doi.org/10.1016/0022-1236(80)90006-3
  51. 51.
    Stone, D.A.: A combinatorial analogue of a theorem of Myers. Ill. J. Math. 20(1), 12–21 (1976)MathSciNetMATHGoogle Scholar
  52. 52.
    Stone, D.A.: Correction to my paper: A combinatorial analogue of a theorem of Myers (Ill. J. Math. 20(1), 12–21 (1976)). Ill. J. Math. 20(3), 551–554 (1976)Google Scholar
  53. 53.
    Sturm, K.T.: On the L p-spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. Anal. 118(2), 442–453 (1993). http://dx.doi.org/10.1006/jfan.1993.1150 MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Sun, L., Yu, X.: Positively curved cubic plane graphs are finite. J. Graph Theory 47(4), 241–274 (2004). http://dx.doi.org/10.1002/jgt.20026 MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Weidmann, J.: Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics, vol. 68. Springer, New York/Berlin (1980). Translated from the German by Joseph SzücsGoogle Scholar
  56. 56.
    Wise, D.T.: Sectional curvature, compact cores, and local quasiconvexity. Geom. Funct. Anal. 14(2), 433–468 (2004). http://dx.doi.org/10.1007/s00039-004-0463-x MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Woess, W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124(3), 385–393 (1998). http://dx.doi.org/10.1017/S0305004197002429 MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Wojciechowski, R.K.: Stochastic completeness of graphs. Thesis (Ph.D.)–City University of New York. ProQuest LLC, Ann Arbor, MI (2008). http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3310649.Google Scholar
  59. 59.
    Zhang, L.: A result on combinatorial curvature for embedded graphs on a surface. Discrete Math. 308(24), 6588–6595 (2008). http://dx.doi.org/10.1016/j.disc.2007.11.007 MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Żuk, A.: On the norms of the random walks on planar graphs. Ann. Inst. Fourier (Grenoble) 47(5), 1463–1490 (1997). http://www.numdam.org/item?id=AIF_1997__47_5_1463_0

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© The Author(s) 2017

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

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