Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Uniformly bounded maximal $\varphi$-disks, Bers space and harmonic maps


Authors: I. Anic, V. Markovic and M. Mateljevic
Journal: Proc. Amer. Math. Soc. 128 (2000), 2947-2956
MSC (1991): Primary 30F30; Secondary 32G15, 58E20
DOI: https://doi.org/10.1090/S0002-9939-00-05325-9
Published electronically: April 7, 2000
MathSciNet review: 1664317
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

We characterize Bers space by means of maximal $\varphi $-disks. As an application we show that the Hopf differential of a quasiregular harmonic map with respect to strongly negatively curved metric belongs to Bers space. Also we give further sufficient or necessary conditions for a holomorphic function to belong to Bers space.


References [Enhancements On Off] (What's this?)

  • 1. Ahlfors, L., Conformal invariants, McGraw-Hill Book Company, 1973. MR 50:10211
  • 2. Ahlfors, L., Lectures on Quasiconformal Mappings, Van Nostrand, 1966. MR 34:336
  • 3. Earle, C.J. and Eells, J., A Fibre bundle description of Teichmüler theory, J. Diff.Geom. 3 (1969) 19-43. MR 43:2737a
  • 4. Gardiner, F.,P., Teichmüller Theory and Quadratic Differentials, New York: Wiley-Interscience Publication, 1987. MR 88m:32044
  • 5. Han, Z-C., Remarks on the geometric behavior of harmonic maps between surfaces, Elliptic and parabolic methods in geometry. Proceedings of a workshop, Minneapolis, May 23-27, 1994, Wellesley. MR 98a:58048
  • 6. Han, Z-C., Tam, L-F., Treibergs, A. and Wan, T., Harmonic maps from the complex plane into surfaces with nonpositive curvature, Commun.Anal.Geom. 3 (1995) 85-114. MR 96k:58057
  • 7. Jost, J., Two-dimensional Geometric Variational Problems, John Wiley & Sons, 1991. MR 92h:58045
  • 8. Lehto, O. and Virtanen, K.I., Quasiconformal Mappings in the Plane, Springer-Verlag, 1973. MR 49:9202
  • 9. Markovic, M. and Mateljevic, M., New versions of Reich-Strebel inequality and uniqueness of harmonic mappings, to appear.
  • 10. Minsky, Y., Harmonic maps, length and energy in Teichmüller space, J. Diff. Geom. 35 (1992), 151-217. MR 93e:58041
  • 11. Pommerenke, Ch., On Bloch functions, J. London Math. Soc. (2), 2 (1970), 689-695. MR 44:1799
  • 12. Reich, E. and Strebel, K., On the Gerstenhaber-Rauch principle, Israel J.Math. 57 (1987) 89-100. MR 88g:30028
  • 13. Schoen, R. and Yau, S.,T., Lectures on Harmonic Maps, Conf. Proc. and Lect. Not. in Geometry and Topology, Vol.II, Inter. Press, 1997. MR 98i:58072
  • 14. Schoen, R. and Yau, S.,T., On univalent harmonic maps between surfaces, Invent.Math. 44 (1978), 265-278. MR 57:17706
  • 15. Stegenga, D. and Stephenson, K., A geometric characterization of analytic functions with bounded mean oscillation, J. London Math. Soc. (2), 24 (1981), 243-254. MR 82m:30036
  • 16. Strebel, K., Quadratic Differentials, Springer-Verlag, 1984. MR 86a:30072
  • 17. Tam, L. and Wan, T., Quasiconformal harmonic diffeomorphism and universal Teichmüler space, J.Diff.Geom. 42 (1995) 368-410. MR 96j:32024
  • 18. Tam, L. and Wan, T., Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials, Comm. Anal. Geom. 4 (1994) 593-625. MR 96m:58057
  • 19. Li, P., Tam, L. and Wang, J., Harmonic diffeomorphisms between hyperbolic Hadamard manifolds, Jour. Geom. Anal., to appear.
  • 20. Wan, T., Constant mean curvature surface, harmonic maps, and universal Teichmüller space, J. Diff. Geom. 35 (1992) 643-657. MR 94a:58053
  • 21. Wolf, M., The Teichmüller theory of harmonic maps, J.Diff.Geom. 29 (1989) 449-479. MR 90h:58023
  • 22. Wolf, M., High-energy degeneration of harmonic maps between surfaces and rays in Teichmüler space, Topology 30 (1991), 517-540. MR 92j:32075

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30F30, 32G15, 58E20

Retrieve articles in all journals with MSC (1991): 30F30, 32G15, 58E20


Additional Information

I. Anic
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski Trg 16, Belgrade, Yugoslavia
Email: ianic@matf.bg.ac.yu

V. Markovic
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: markovic@math.umn.edu

M. Mateljevic
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski Trg 16, Belgrade, Yugoslavia
Email: miodrag@matf.bg.ac.yu

DOI: https://doi.org/10.1090/S0002-9939-00-05325-9
Keywords: Quadratic differentials, Bers space, quasiregular harmonic maps, negatively curved metrics
Received by editor(s): April 20, 1998
Received by editor(s) in revised form: August 27, 1998, and November 18, 1998
Published electronically: April 7, 2000
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society