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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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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