Convergence of harmonic maps on the Poincaré disk
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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.References
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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
- MR Author ID: 711172
- Email: wallgreat@lycos.com
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2483-2493
- MSC (2000): Primary 58E20; Secondary 30C62
- DOI: https://doi.org/10.1090/S0002-9939-04-07465-9
- MathSciNet review: 2052429