Mathematische Annalen

, Volume 364, Issue 3–4, pp 793–811 | Cite as

Proper harmonic maps between asymptotically hyperbolic manifolds

  • Kazuo Akutagawa
  • Yoshihiko Matsumoto


Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with \(C^1\) boundary conditions at infinity between asymptotically hyperbolic manifolds.

Mathematics Subject Classification

53C43 58J32 



The authors would like to thank Robin Graham and Man Chun Leung for their help during preparation of this article. They would also like to thank the anonymous referee for valuable comments. The second author would like to acknowledge the hospitality of the École normale supérieure, where he was visiting while the revision of the manuscript was carried out. The first author is supported in part by the Grant-in-Aid for Challenging Exploratory Research, Japan Society for the Promotion of Science, No. 24654009. The second author is supported in part by the Grant-in-Aid for JSPS Fellows, Japan Society for the Promotion of Science, No. 26-11754.


  1. 1.
    Akutagawa, K.: Harmonic diffeomorphisms of the hyperbolic plane. Trans. Am. Math. Soc. 342(1), 325–342 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque (265), vi+109 (2000)Google Scholar
  3. 3.
    Cheng, S.Y.: Liouville theorem for harmonic maps. In: Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., vol. XXXVI, pp. 147–151. Amer. Math. Soc., Providence (1980)Google Scholar
  4. 4.
    Chruściel, P.T., Delay, E., Lee, J.M., Skinner, D.N.: Boundary regularity of conformally compact Einstein metrics. J. Differ. Geom. 69(1), 111–136 (2005)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ding, W.Y., Wang, Y.D.: Harmonic maps of complete noncompact Riemannian manifolds. Int. J. Math. 2(6), 617–633 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Donnelly, H.: Asymptotic Dirichlet problem for harmonic maps with bounded image. In: Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), vol. 91, pp. 1–6 (2002)Google Scholar
  7. 7.
    Economakis, M.: Boundary regularity of the harmonic map problem between asymptotically hyperbolic manifolds. Thesis (Ph.D.), University of Washington. ProQuest LLC, Ann Arbor (1993)Google Scholar
  8. 8.
    Economakis, M.: A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces. J. Geom. Anal. 3(1), 27–36 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eells Jr, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fefferman, C., Graham, C.R.: \(Q\)-curvature and Poincaré metrics. Math. Res. Lett. 9(2–3), 139–151 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fotiadis, A.: Harmonic maps between noncompact manifolds. J. Nonlinear Math. Phys. 15(suppl. 3), 176–184 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, vol. 224, 2nd edn. Springer, Berlin (1983)Google Scholar
  13. 13.
    Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152(1), 89–118 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hamilton, R.S.: Harmonic maps of manifolds with boundary. Lecture Notes in Mathematics, vol. 471. Springer, Berlin (1975)Google Scholar
  15. 15.
    Jäger, W., Kaul, H.: Uniqueness of harmonic mappings and of solutions of elliptic equations on Riemannian manifolds. Math. Ann. 240(3), 231–250 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jost, J.: Harmonic mappings between Riemannian manifolds. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 4. Australian National University, Centre for Mathematical Analysis, Canberra (1984)Google Scholar
  17. 17.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lee, J.M.: Fredholm operators and Einstein metrics on conformally compact manifolds. Mem. Am. Math. Soc. 183(864), vi+83 (2006)Google Scholar
  19. 19.
    Leung, M.C.: Harmonic maps between asymptotically hyperbolic spaces. Thesis (Ph.D.), University of Michigan. ProQuest LLC, Ann Arbor (1991)Google Scholar
  20. 20.
    Li, P., Tam, L.F.: The heat equation and harmonic maps of complete manifolds. Invent. Math. 105(1), 1–46 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, P., Tam, L.F.: Uniqueness and regularity of proper harmonic maps. Ann. Math. (2) 137(1), 167–201 (1993)Google Scholar
  22. 22.
    Li, P., Tam, L.F.: Uniqueness and regularity of proper harmonic maps. II. Indiana Univ. Math. J. 42(2), 591–635 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mazzeo, R.: The Hodge cohomology of a conformally compact metric. J. Differ. Geom. 28(2), 309–339 (1988)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mazzeo, R.R., Melrose, R.B.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75(2), 260–310 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schoen, R., Yau, S.T.: Compact group actions and the topology of manifolds with nonpositive curvature. Topology 18(4), 361–380 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Institute of TechnologyTokyoJapan

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