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The Journal of Geometric Analysis

, Volume 26, Issue 2, pp 750–781 | Cite as

Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

  • Gerhard KnieperEmail author
  • Norbert Peyerimhoff
Article
  • 213 Downloads

Abstract

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature \(h\). In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds \(X\) with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon–Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification \(\overline{X}\).

Keywords

Asymptotically harmonic manifolds Harmonic functions  Visibility measures Gromov hyperbolicity Dirichlet problem at infinity Mean value property at infinity 

Mathematics Subject Classification

Primary 53C25 Secondary 37D20 53C23 53C40 

Notes

Acknowledgments

The authors would like to thank Evangelia Samiou for bringing the references [10, 11] to their attention.

References

  1. 1.
    Ancona, A.: Positive harmonic functions and hyperbolicity, In: J. Král et al. (eds.) Potential Theory-Surveys and Problems (Prague, 1987). Lecture Notes in Mathematics, vol. 1344, pp. 1–23. Springer, Berlin (1988)Google Scholar
  2. 2.
    Ancona, A.: Théorie du potentiel sur les graphes et les variétés, In: École d’été de Probabilités de Saint-Flour XVIII–1988. Lecture Notes in Mathematics, vol. 1427, pp. 1–112. Springer, Berlin (1990)Google Scholar
  3. 3.
    Ballmann, W., Brin, M., Eberlein, P.: Structure of manifolds of nonpositive curvature I. Ann. Math. 122(1), 171–203 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ballmann, W.: On the Dirichlet problem at infinity for manifolds of nonpositive curvature. Forum Math. 1(2), 201–213 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Buyalo, S., Schroeder, V.: Elements of Asymptotic Geometry. European Mathematical Society (EMS), Zürich (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Castillon, P., Sambusetti, A.: On asymptotically harmonic manifolds of negative curvature. Math. Z. doi: 10.1007/s00209-014-1293-7, 18 March 2014. See also arXiv:1203.2482, 12 March (2012)
  7. 7.
    Coornaert, M., Delzant, T., Papadopoulos, A.: Géométrie et théorie des groupes. In: Lecture Notes in Mathematics, vol. 1441. Springer, Berlin (1990)Google Scholar
  8. 8.
    Eberlein, P., O’Neill, B.: Visibility manifolds. Pac. J. Math. 46, 45–109 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Friedrich, Th: Die Fisher-Information und symplektische Strukturen. Math. Nachr. 153, 273–296 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Itoh, M., Satoh, H.: Information geometry of Poisson kernels on Damek–Ricci spaces. Tokyo J. Math. 33, 129–144 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Itoh, M., Satoh, H.: Fisher information geometry, Poisson kernel and asymptotical harmonicity. Differ. Geom. Appl. 29(suppl. 1), S107–S115 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karp, L., Peyerimhoff, N.: Horospherical means and uniform distribution of curves of constant geodesic curvature. Math. Z. 231, 655–677 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Knieper, G.: Dynamics, hyperbolic, geometry, Riemannian, In: Hasselblatt, B., Katok, A. (eds.) Handbook of Dynamical Systems, vol. 1A, pp. 453–545. Elsevier, Amsterdam (2002)Google Scholar
  14. 14.
    Knieper, G.: New results on noncompact harmonic manifolds. Comment. Math. Helv. 87, 669–703 (2012). arXiv:0910.3872, 20 October (2009)
  15. 15.
    Knieper, G., Peyerimhoff, N.: Noncompact harmonic manifolds. Oberwolfach Preprint OWP 2013–08. arXiv:1302.3841, 15 February (2013)
  16. 16.
    Knieper, G., Peyerimhoff, N.: Geometric properties of rank one asymptotically harmonic manifolds. J. Differ. Geom. (2014) (To appear) arXiv:1307.0629, 7 January (2014)
  17. 17.
    Ledrappier, F.: Harmonic measures and Bowen–Margulis measures. Israel J. Math. 71(3), 275–287 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Peyerimhoff, N., Samiou, E.: Integral geometric properties of non-compact harmonic spaces. J. Geom. Anal. 25(1), 122–148 (2015)Google Scholar
  19. 19.
    Rouvière, F.: Espaces de Damek–Ricci, géometrie et analyse. Séminaires & Congrès 7, 45–100 (2003)zbMATHGoogle Scholar
  20. 20.
    Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zimmer, A.M.: Compact asymptotically harmonic manifolds. J. Mod. Dyn. 6(3), 377–403. arXiv:1205.2271, 16 October (2012)
  22. 22.
    Zimmer, A.M.: Boundaries of non-compact harmonic manifolds. Geom. Dedicata 168, 339–357 (2014). arXiv:1208.4802, 16 December (2012)

Copyright information

© Mathematica Josephina, Inc. 2015

Authors and Affiliations

  1. 1.Faculty of MathematicsRuhr University BochumBochumGermany
  2. 2.Department of Mathematical SciencesDurham UniversityDurhamUK

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