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Convergence of harmonic maps on the Poincaré disk

Author: Guowu Yao
Journal: Proc. Amer. Math. Soc. 132 (2004), 2483-2493
MSC (2000): Primary 58E20; Secondary 30C62
Published electronically: March 3, 2004
MathSciNet review: 2052429
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Abstract: Let $\{f_n:{\mathbb D}\to{ \mathbb D}\}$ be a sequence of locally quasiconformal harmonic maps on the unit disk ${\mathbb D}$ with respect to the Poincaré metric. Suppose that the energy densities of $f_n$ are uniformly bounded from below by a positive constant and locally uniformly bounded from above. Then there is a subsequence of $\{f_n\}$ that locally uniformly converges on ${\mathbb D}$, and the limit function is either a locally quasiconformal harmonic map of the Poincaré disk or a constant. Especially, if the limit function is not a constant, the subsequence can be chosen to satisfy some stronger conditions. As an application, it is proved that every point of the space $T_0({\mathbb D})$, a subspace of the universal Teichmüller space, can be represented by a quasiconformal harmonic map that is an asymptotic hyperbolic isometry.

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  • 1. I. Anic, V. Markovic, and M. Mateljievic, Uniformly bounded maximal $\phi$-disks, Bers space and harmonic maps, Proc. Amer. Math. Soc., 128 (2000), 2947-2956. MR 2000m:30054
  • 2. S. Y. Cheng, L. F. Tam, and Tom Y. H. Wan, Harmonic maps with finite total energy, Proc. Amer. Math. Soc., 124 (1996), 275-284. MR 96d:58031
  • 3. A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math., 157 (1986), 23-48. MR 87j:30041
  • 4. C. J. Earle, V. Markovic, and D. Saric, Barycentric extension and the Bers embedding for asymptotic Teichmüller space, Contemp. Math. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 87-105. MR 2003i:30072
  • 5. R. Fehlmann, Ueber extremale quasikonforme Abbildungen, Comment. Math. Helv., 56 (1981), 558-580. MR 83e:30024
  • 6. F. P. Gardiner and D. P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math., 114 (1992), 683-736. MR 95h:30020
  • 7. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Springer-Verlag, 1983. MR 86c:35035
  • 8. O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, Berlin and New York, 1973. MR 49:9202
  • 9. Z. Li, On the boundary value problem for harmonic maps of the Poincaré disc, Chinese Sci. Bull., 42 (1997), 2025-2045. MR 2000b:30060
  • 10. V. Markovic, Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces, J. London Math. Soc. (2) 65 (2002), 103-114. MR 2002k:32015
  • 11. R. Schoen, The role of harmonic mappings in rigidity and deformation problems, Collection: Complex Geometry (Osaka, 1990), Lecture Notes in Pure and Applied Mathematics, Vol. 143, Dekker, New York, 1993, pp. 179-200. MR 94g:58055
  • 12. R. Schoen and S. T. Yau, On univalent harmonic maps between surfaces, Invent. Math., 44 (1978), 265-278. MR 57:17706
  • 13. K. Strebel, On the existence of extremal Teichmüller mappings, J. Analyse Math., 30 (1976), 464-480. MR 55:12912
  • 14. L. F. Tam and Tom Y. H. Wan, Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials, Comm. Anal. Geom., 2 (1994), 593-625. MR 96m:58057
  • 15. L. F. Tam and Tom Y. H. Wan, Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space, J. Differential Geom. 42 (1995), 368-410. MR 96j:32024
  • 16. Tom Y. H. Wan, Constant mean curvature surface, harmonic maps and universal Teichmüller space, J. Differential Geom., 35 (1992), 643-657. MR 94a:58053
  • 17. M. Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), 449-479. MR 90h:58023
  • 18. G. W. Yao, Harmonic maps and asymptotic Teichmüller space, preprint (2001).

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Additional Information

Guowu Yao
Affiliation: School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
Address at time of publication: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

Keywords: Harmonic map, energy density, locally quasiconformal map, asymptotic hyperbolic isometry
Received by editor(s): May 17, 2002
Received by editor(s) in revised form: May 15, 2003
Published electronically: March 3, 2004
Additional Notes: This research was supported by the “973” Project Foundation of China (Grant No. TG199075105) and the Foundation for Doctoral Programme
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2004 American Mathematical Society

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