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



$\overline{\partial }$-energy integral and harmonic mappings

Author: Guowu Yao
Journal: Proc. Amer. Math. Soc. 131 (2003), 2271-2277
MSC (2000): Primary 58E20; Secondary 30C62
Published electronically: October 24, 2002
MathSciNet review: 1963777
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Abstract: In this paper, we discuss harmonic mappings on the unit disk with respect to any metric by using the $\overline{\partial }$-energy integral that was first introduced by Li in 1997 to treat quasiconformal harmonic mappings on the Poincaré disk, instead of the total energy integral. Some basic properties of harmonic mappings are given. Moreover, we give a new proof of the uniqueness theorem of Markovic and Mateljevic, which is more explicit and natural.

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  • [Ah] L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York, 1966. MR 34:336
  • [J] J. Jost, Harmonic Maps between Surfaces, Lecture Notes in Mathematics, vol. 1062, Springer-Varlag, 1984. MR 85j:58046
  • [LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, 1973. MR 49:9202
  • [Li] Z. Li, On the boundary value problem for harmonic maps of the Poincaré disc, Chinese Science Bulletin 42 (1997), 2025-2045. MR 2000b:30060
  • [LT] P. Li and L.-F. Tam, Uniqueness and regularity of proper harmonic maps II, Indiana University Math. Journal 42 (1993), 591-635. MR 94i:58044
  • [MM] V. Markovic and M. Mateljevic, New version of the main inequality and the uniqueness of harmonic maps, Journal d'Analyse Math. 79 (1999), 315-334. MR 2001d:30027
  • [Re] E. Reich, On the variational principle of Gerstenhaber and Rauch, Ann. Acad. Sci. Fenn., Ser. A. I. Math. 10 (1985), 469-475. MR 87b:30036
  • [RS] E. Reich and K. Strebel, On the Gerstenhaber-Rauch principle, Israel J. Math. 57 (1987), 89-100. MR 88g:30028
  • [RS1] E. Reich and K. Strebel, On the quasiconformal mappings which keep the boundary points fixed, Trans. Amer. Math. Soc. 138 (1969), 211-222. MR 38:6059
  • [RS2] E. Reich and K. Strebel, Extremal quasiconformal mappings with given boundary values, A collection of Papers dedicated to Lipman Bers, Contributions to Analysis, Academic Press, 1974, pp. 375-392. MR 50:13511
  • [S] J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. École. Norm. Sup. 4 (1978), 211-228. MR 80b:58031
  • [SY] R. Schoen and S. T. Yau, On the univalent harmonic maps between surfaces, Invent. Math. 44 (1978), 265-278. MR 57:17706
  • [We] Wei H. B., On the uniqueness problem of harmonic mappings, Proc. Amer. Math. Soc. 124 (1996), 2337-2341. MR 96j:30032
  • [Ya] G. W. Yao, Improved Reich-Strebel inequality and harmonic mappings, submitted to J. Anal. Math. (2001).

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Additional Information

Guowu Yao
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Keywords: Harmonic mapping, Main Inequality
Received by editor(s): February 20, 2002
Published electronically: October 24, 2002
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2002 American Mathematical Society