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

DOI:
https://doi.org/10.1090/S0002-9939-04-07465-9

Published electronically:
March 3, 2004

MathSciNet review:
2052429

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a sequence of locally quasiconformal harmonic maps on the unit disk with respect to the Poincaré metric. Suppose that the energy densities of are uniformly bounded from below by a positive constant and locally uniformly bounded from above. Then there is a subsequence of that locally uniformly converges on , 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 , a subspace of the universal Teichmüller space, can be represented by a quasiconformal harmonic map that is an asymptotic hyperbolic isometry.

**1.**I. Anic, V. Markovic, and M. Mateljievic,*Uniformly bounded maximal -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).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
58E20,
30C62

Retrieve articles in all journals with MSC (2000): 58E20, 30C62

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

Email:
wallgreat@lycos.com

DOI:
https://doi.org/10.1090/S0002-9939-04-07465-9

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