Skip to main content
SpringerLink
Log in

Polynomial growth harmonic functions on finitely generated abelian groups

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

In the present paper, we develop geometric analysis techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We provide a geometric analysis proof of the classical Heilbronn theorem (Heilbronn in Proc Camb Philos Soc 45:194–206, 1949) and the recent Nayar theorem (Nayar in Bull Pol Acad Sci Math 57:231–242, 2009) on polynomial growth harmonic functions on lattices \(\mathbb Z ^n\) that does not use a representation formula for harmonic functions. In the abelian group case, by Yau’s gradient estimate, we actually give a simplified proof of a general polynomial growth harmonic function theorem of (Alexopoulos in Ann Probab 30:723–801, 2002). We calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups by linear algebra, rather than by Floquet theory Kuchment and Pinchover (Trans Am Math Soc 359:5777–5815, 2007). While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself. Moreover, we also calculate the dimension of solutions to higher order Laplace operators.

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

References

  1. Alexopoulos, G.: An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Canad. J. Math. 44(4), 691–727 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexopoulos, G.: An application of homogenization theory to harmonic analysis on solvable Lie groups of polynomial growth. Pacific J. Math. 159(1), 19–45 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alexopoulos, G.: Random walks on discrete groups of polynomial volume growth. Ann. Probab. 30(2), 723–801 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alexopoulos, G., Lohoué, N.: Sobolev inequalities and harmonic functions of polynomial growth. J. London Math. Soc. (2) 48(3), 452–464 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Avellaneda, M., Lin, F.H.: Un théorème de Liouville pour des équations elliptiques à coefficients périodiques. C. R. Acad. Sci. Paris Sér. I Math. 309(5), 245–250 (1989)

  6. Burago, D., Burago, Yu., Ivanov, S.: A course in metric geometry, Graduate Studies in Mathematics 33. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  7. Chen, R., Wang, J.: Polynomial growth solutions to higher-order linear elliptic equations and systems. Pacific J. Math. 229(1), 49–61 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28, 333–354 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chung, F.R.K., Yau, S.T.: Logarithmic Harnack inequalities. Math. Res. Lett. 3(6), 793–812 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Colding, T.H., Minicozzi II, W.P.: Harmonic functions with polynomial growth. J. Diff. Geom. 46(1), 1–77 (1997)

    MathSciNet  MATH  Google Scholar 

  11. Colding, T.H., Minicozzi II, W.P.: Harmonic functions on manifolds. Ann. of Math. (2) 146(3), 725–747 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Colding, T.H., Minicozzi II, W.P.: Weyl type bounds for harmonic functions. Invent. Math. 131(2), 257–298 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colding, T.H., Minicozzi II, W.P.: Liouville theorems for harmonic sections and applications. Comm. Pure Appl. Math. 51(2), 113–138 (1998)

    Article  MathSciNet  Google Scholar 

  14. Coulhon, T., Grigoryan, A.: Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8(4), 656–701 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Coulhon, T., Saloff-Coste, L.: Variétés riemanniennes isométriques à l’infini. Rev. Mat. Iberoamericana 11(3), 687–726 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Delmotte, T.: Harnack inequalities on graphs, Séminaire de Théorie Spectrale et Géométrie, 16, 217–228 (1997–1998)

  17. Delmotte, T.: Inégalité de Harnack elliptique sur les graphes. Colloq. Math. 72(1), 19–37 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Donnelly, H., Fefferman, C.: Nodal domains and growth of harmonic functions on noncompact manifolds. J. Geom. Anal. 2(1), 79–93 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Duffin, R.J., Shelly, E.P.: Difference equations of polyharmonic type. Duke Math. J. 25, 209–238 (1958)

    Article  MathSciNet  Google Scholar 

  20. Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Éudes Sci. Publ. Math. No. 53, 53–73 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Holopainen, I., Soardi, P.M.: A strong Liouville theorem for p-harmonic functions on graphs. Ann. Acad. Sci. Fenn. Math. 22(1), 205–226 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Heilbronn, H.A.: On discrete harmonic functions. Proc. Cambridge Philos. Soc. 45, 194–206 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hobson, E.W.: The theory of spherical and ellipsoidal harmonics. Chelsea Publishing Company, New York (1955)

  24. Hua, B.: Generalized Liouville theorem in nonnegatively curved Alexandrov spaces. Chin. Ann. Math. Ser. B 30(2), 111–128 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hua, B.: Harmonic functions of polynomial growth on singular spaces with nonnegative Ricci curvature. Proc. Amer. Math. Soc. 139, 2191–2205 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hua, B. Jost, J.: Polynomial growth harmonic functions on groups of polynomial volume growth, preprint, arXiv:1201.5238.

  27. Hua, B., Jost, J., Liu S.: Geometric aspects of infinite semiplanar graphs with nonnegative curvature, preprint, arXiv:1107.2826.

  28. Kim, S.W., Lee, Y.H.: Polynomial growth harmonic functions on connected sums of complete Riemannian manifolds. Math. Z. 233(1), 103–113 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kleiner, B.: A new proof of Gromov’s theorem on groups of polynomial growth. J. Amer. Math. Soc. 23(3), 815–829 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kuchment, P., Pinchover, Y.: Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds. Trans. Amer. Math. Soc. 359(12), 5777–5815 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Milnor, J.: A note on curvature and fundamental group. J. Differential Geometry 2, 1–7 (1968)

    MathSciNet  MATH  Google Scholar 

  32. Moser, J., Struwe, M.: On a Liouville-type theorem for linear and nonlinear elliptic differential equations on a torus. Bol. Soc. Brasil. Mat. (N.S.) 23(1—-2), 1–20 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lawler, G.F.: Intersections of random walks. Probability and its Applications. Birkhäuser Boston Inc., Boston, MA (1991)

    Book  MATH  Google Scholar 

  34. Lee, Y.H.: Polynomial growth harmonic functions on complete Riemannian manifolds. Rev. Mat. Iberoamericana 20(2), 315–332 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, P.: Harmonic sections of polynomial growth. Math. Res. Lett. 4(1), 35–44 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, P.: Harmonic functions and applications to complete manifolds (lecture notes), preprint.

  37. Li, P., Tam, L.-F.: Complete surfaces with finite total curvature. J. Differential Geom. 33(1), 139–168 (1991)

    MathSciNet  MATH  Google Scholar 

  38. Li, P., Wang, J.: Mean value inequalities. Indiana Univ. Math. J. 48(4), 1257–1283 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, P., Wang, J.: Counting dimensions of L-harmonic functions. Ann. of Math. (2) 152(2), 645–658 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, P., Wang, J.: Polynomial growth solutions of uniformly elliptic operators of non-divergence form. Proc. Amer. Math. Soc. 129(12), 3691–3699 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lin, F.: Asymptotically conic elliptic operators and Liouville type theorems, Geometric analysis and the calculus of variations, pp. 217–238. Int. Press, Cambridge, MA (1996)

    Google Scholar 

  42. Lin, Y., Xi, L.: Lipschitz property of harmonic function on graphs. J. Math. Anal. Appl. 366(2), 673–678 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lin, Y., Yau, S.T.: Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett. 17(2), 343–356 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Nayar, P.: On polynomially bounded harmonic functions on the \(\mathbb{Z}^d\) lattice. Bull. Pol. Acad. Sci. Math., 57(3-4), 231-242 (2009)

    Google Scholar 

  45. Raghunathan, M.S.: Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. Springer-Verlag, New York-Heidelberg (1972)

    Book  Google Scholar 

  46. Robinson, D.J.S.: A course in the theory of groups, second edition, Graduate Texts in Mathematics, 80. Springer-Verlag, New York (1996)

    Book  MATH  Google Scholar 

  47. Shalom, Y., Tao, T.: A finitary version of Gromov’s polynomial growth theorem. Geom. Funct. Anal. 20(6), 1502–1547 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sung, C.-J., Tam, L.-F., Wang, J.: Spaces of harmonic functions. J. London Math. Soc. (2) 61(3), 789–806 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. Suzuki, M.: Group theory I, Translated from the Japanese by the author, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 247. Springer-Verlag, Berlin-New York (1982)

    Google Scholar 

  50. Tam, L.-F.: A note on harmonic forms on complete manifolds. Proc. Amer. Math. Soc. 126(10), 3097–3108 (1998)

    Article  MathSciNet  Google Scholar 

  51. Wang, J.: Linear growth harmonic functions on complete manifolds. Comm. Anal. Geom. 4, 683–698 (1995)

    Google Scholar 

  52. Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yau, S.T.: Enseign. Math. Nonlinear analysis in geometry. 33(2), 109–158 (1987)

    MATH  Google Scholar 

  54. Yau, S. T.: Differential Geometry: Partial Differential Equations on Manifolds, Proc. of Symposia in Pure Mathematics, 54, part 1, Ed. by R.Greene and S.T. Yau (1993)

Download references

Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, 04103 , Leipzig, Germany

    Bobo Hua, Jürgen Jost & Xianqing Li-Jost

Authors
  1. Bobo Hua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürgen Jost
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xianqing Li-Jost
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bobo Hua.

Additional information

J. Jost is supported by ERC Advanced Grant FP7-267087.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hua, B., Jost, J. & Li-Jost, X. Polynomial growth harmonic functions on finitely generated abelian groups. Ann Glob Anal Geom 44, 417–432 (2013). https://doi.org/10.1007/s10455-013-9374-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-013-9374-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation