Proper harmonic maps between asymptotically hyperbolic manifolds
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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 Classification53C43 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.
- 2.Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque (265), vi+109 (2000)Google Scholar
- 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
- 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.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
- 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
- 14.Hamilton, R.S.: Harmonic maps of manifolds with boundary. Lecture Notes in Mathematics, vol. 471. Springer, Berlin (1975)Google Scholar
- 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
- 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.Leung, M.C.: Harmonic maps between asymptotically hyperbolic spaces. Thesis (Ph.D.), University of Michigan. ProQuest LLC, Ann Arbor (1991)Google Scholar
- 21.Li, P., Tam, L.F.: Uniqueness and regularity of proper harmonic maps. Ann. Math. (2) 137(1), 167–201 (1993)Google Scholar