$\overline {\partial }$-energy integral and harmonic mappings
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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 Marković and Mateljević, which is more explicit and natural.References
- Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
- Jürgen Jost, Harmonic maps between surfaces, Lecture Notes in Mathematics, vol. 1062, Springer-Verlag, Berlin, 1984. MR 754769, DOI 10.1007/BFb0100160
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463
- Zhong Li, On the boundary value problem for harmonic maps of the Poincaré disc, Chinese Sci. Bull. 42 (1997), no. 24, 2025–2045. MR 1641041, DOI 10.1007/BF02882940
- Peter Li and Luen-Fai Tam, Uniqueness and regularity of proper harmonic maps. II, Indiana Univ. Math. J. 42 (1993), no. 2, 591–635. MR 1237061, DOI 10.1512/iumj.1993.42.42027
- V. Marković and M. Mateljević, A new version of the main inequality and the uniqueness of harmonic maps, J. Anal. Math. 79 (1999), 315–334. MR 1749316, DOI 10.1007/BF02788245
- Edgar Reich, On the variational principle of Gerstenhaber and Rauch, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 469–475. MR 802510, DOI 10.5186/aasfm.1985.1052
- Edgar Reich and Kurt Strebel, On the Gerstenhaber-Rauch principle, Israel J. Math. 57 (1987), no. 1, 89–100. MR 882248, DOI 10.1007/BF02769462
- Edgar Reich and Kurt Strebel, On quasiconformal mappings which keep the boundary points fixed, Trans. Amer. Math. Soc. 138 (1969), 211–222. MR 237778, DOI 10.1090/S0002-9947-1969-0237778-3
- Edgar Reich and Kurt Strebel, Extremal quasiconformal mappings with given boundary values, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 375–391. MR 0361065
- J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 211–228. MR 510549
- Richard Schoen and Shing Tung Yau, On univalent harmonic maps between surfaces, Invent. Math. 44 (1978), no. 3, 265–278. MR 478219, DOI 10.1007/BF01403164
- Hanbai Wei, On the uniqueness problem of harmonic quasiconformal mappings, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2337–2341. MR 1307523, DOI 10.1090/S0002-9939-96-03178-4
- G. W. Yao, Improved Reich-Strebel inequality and harmonic mappings, submitted to J. Anal. Math. (2001).
Additional Information
- Guowu Yao
- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- MR Author ID: 711172
- Email: wallgreat@lycos.com
- Received by editor(s): February 20, 2002
- Published electronically: October 24, 2002
- Communicated by: Juha M. Heinonen
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 2271-2277
- MSC (2000): Primary 58E20; Secondary 30C62
- DOI: https://doi.org/10.1090/S0002-9939-02-06757-6
- MathSciNet review: 1963777