Skip to main content
SpringerLink
Log in

The Kazdan–Warner equation on canonically compactifiable graphs

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We study the Kazdan–Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Here, countable means finite or countably infinite, although we are of course mainly interested in the countably infinite case.

References

  1. Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D., Yau, S.-T.: Li-Yau inequality on graphs. J. Differ. Geom. 99(3), 359–405 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauer, F., Hua, B., Yau, S.-T.: Davies–Gaffney–Grigor’yan lemma on graphs. Commun. Anal. Geom. 23(5), 1031–1068 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauer, F., Keller, M., Wojciechowski, R.K.: Cheeger inequalities for unbounded graph Laplacians. J. Eur. Math. Soc. (JEMS) 17(2), 259–271 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25(1), 141–159 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243–263 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, W.X., Li, C.: Qualitative properties of solutions to some nonlinear elliptic equations in \(\mathbf{R}^2\). Duke Math. J. 71(2), 427–439 (1993)

    Article  MathSciNet  Google Scholar 

  7. Ding, W., Jost, J., Li, J., Wang, G.: The differential equation \(\Delta u=8\pi -8\pi he^u\) on a compact Riemann surface. Asian J. Math. 1(2), 230–248 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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). Dedicated to the memory of Jürgen K. Moser

  9. Erbar, M., Maas, J.: Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206(3), 997–1038 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Folz, M.: Volume growth and stochastic completeness of graphs. Trans. Am. Math. Soc. 366(4), 2089–2119 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Forman, R.: Bochner’s method for cell complexes and combinatorial Ricci curvature. Discrete Comput. Geom. 29(3), 323–374 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Volume 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (2011)

    Google Scholar 

  13. Ge, H.: Kazdan–Warner equation on graph in the negative case. J. Math. Anal. Appl. 453(2), 1022–1027 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Georgakopoulos, A., Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.K.: Graphs of finite measure. J. Math. Pures Appl. (9) 103(5), 1093–1131 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ge, H., Jiang, W.: Kazdan–Warner Equation on Infinite Graphs, Preprint. arXiv:1706.08698 (2017)

  16. Grigor’yan, A., Lin, Y., Yang, Y.: Kazdan-Warner equation on graph. Calc. Var. Partial Differ. Equ. 55(4), 92 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Golénia, S.: Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians. J. Funct. Anal. 266(5), 2662–2688 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hua, B., Jost, J., Liu, S.: Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature. J. Reine Angew. Math. 700, 1–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hua, B., Keller, M.: Harmonic functions of general graph Laplacians. Calc. Var. Partial Differ. Equ. 51(1–2), 343–362 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.: Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory 2(4), 397–432 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hua, B., Lin, Y.: Stochastic completeness for graphs with curvature dimension conditions. Adv. Math. 306, 279–302 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hua, B., Mugnolo, D.: Time regularity and long-time behavior of parabolic \(p\)-Laplace equations on infinite graphs. J. Differ. Equ. 259(11), 6162–6190 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. 666, 189–223 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Keller, M., Lenz, D., Schmidt, M., Wirth, M.: Diffusion determines the recurrent graph. Adv. Math. 269, 364–398 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Keller, M., Lenz, D., Schmidt, M., Wojciechowski, R.K.: Note on uniformly transient graphs. Rev. Math. Iberoam. 33(3), 831–860 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Keller, M., Mugnolo, D.: General Cheeger inequalities for \(p\)-Laplacians on graphs. Nonlinear Anal. 147, 80–95 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Keller, M., Pinchover, Y., Pogorzelski, F.: Optimal Hardy inequalities for Schrödinger operators on graphs. Commun. Math. Phys. 358(2), 767–790 (2018)

    Article  MATH  Google Scholar 

  28. Keller, M., Peyerimhoff, N., Pogorzelski, F.: Sectional curvature of polygonal complexes with planar substructures. Adv. Math. 307, 1070–1107 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kazdan, J.L., Warner, F.W.: Curvature functions for compact \(2\)-manifolds. Ann. Math. 2(99), 14–47 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kazdan, J.L., Warner, F.W.: Curvature functions for open \(2\)-manifolds. Ann. Math. 2(99), 203–219 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mugnolo, D.: Parabolic theory of the discrete \(p\)-Laplace operator. Nonlinear Anal. 87, 33–60 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Münch, F.: Li-Yau Inequality on Finite Graphs Via Non-linear Curvature Dimension Conditions, Preprint. arXiv:1412.3340F (2013)

  33. Ollivier, Y.: Ricci curvature of metric spaces. C. R. Math. Acad. Sci. Paris 345(11), 643–646 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256(3), 810–864 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Stollmann, P.: Scattering by obstacles of finite capacity. J. Funct. Anal. 121(2), 416–425 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph. J. Math. Anal. Appl. 370(1), 146–158 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wise, D.T.: Nonpositive immersions, sectional curvature, and subgroup properties. Electron. Res. Announc. Am. Math. Soc. 9, 1–9 (2003). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wojciechowski, R.K.: Stochastic completeness of graphs. ProQuest LLC, Ann Arbor, MI (2008)

    Google Scholar 

  39. Zeidler, E.: Applied Functional Analysis, Volume 109 of Applied Mathematical Sciences. Springer-Verlag, New York (1995).

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Potsdam, 14476, Potsdam, Germany

    Matthias Keller & Michael Schwarz

Authors
  1. Matthias Keller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Schwarz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Keller.

Additional information

Communicated by J. Jost.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keller, M., Schwarz, M. The Kazdan–Warner equation on canonically compactifiable graphs. Calc. Var. 57, 70 (2018). https://doi.org/10.1007/s00526-018-1329-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-018-1329-7

Mathematics Subject Classification

Search

Navigation