Abstract
Let \(\{f_n:\mathbb {D}\rightarrow \mathbb {D}\}\) be a sequence of K-quasiconformal harmonic maps on the unit disk \(\mathbb {D}\) with respect to the Poincaré metric. It is known that there is a subsequence of \(\{f_n\}\) that uniformly converges on \(\overline{\mathbb {D}}\), and the limit function is either a K-quasiconformal harmonic map of the Poincaré disk or a constant. In this paper, it is shown that, if the limit function is not a constant, the subsequence can be chosen such that its derivative sequence of arbitrary order uniformly converges to the corresponding derivative of the limit function on compact subsets of \(\mathbb {D}\).
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References
Douady, A., Earle, C.J.: Conformally natural extension of homeomorphisms of the circle. Acta Math. 157, 23–48 (1986)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd edn. Springer, New York (1983)
Li, P., Tam, L.F.: Uniqueness and regularity of proper harmonic maps. Indiana Univ. Math. J. 42, 591–635 (1993)
Lehto, O., Virtanen, K.I.: Quasiconformal Mapping in the Plane. Springer, Berlin (1973)
Marković, V.: Harmonic diffeomorphism of noncompact surfaces and Teichmüller spaces. J. Lond. Math. Soc. 65, 103–114 (2002)
Marković, V.: Harmonic maps and the Schoen conjecture. http://www.its.caltech.edu/~markovic (preprint)
Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Sci. École Norm. Supérieure 11, 211–228 (1978)
Schoen, R.: The role of harmonic mappings in rigidity and deformation problems. In: Collection: Complex Geometry (Osaka, 1990). Lecture Notes in Pure and Applied Mathematics, vol. 143, pp. 179–200. Dekker, New York (1993)
Schoen, R., Yau, S.T.: On univalent harmonic maps between surfaces. Invent. Math. 44, 265–278 (1978)
Tam, L.F., Tom, Y.H.: Wan, Harmonic diffeomorphisms into Cartan–Hadamard surfaces with prescribed Hopf differentials. Commun. Anal. Geom. 2, 593–625 (1994)
Tam, L.F., Wan, T.Y.H.: Quasiconformal harmonic diffeomorphism and the universal Teichmüller sapce. J. Differ. Geom. 42, 368–410 (1995)
Wan, T.Y.H.: Constant mean curvature surface, harmonic maps and universal Teichmüller space. J. Differ. Geom. 35, 643–657 (1992)
Wolf, M.: The Teichmüller theory of harmonic maps. J. Differ. Geom. 29, 449–479 (1989)
Yao, G.W.: \(\bar{\partial }\)-Energy integral and harmonic mappings. Proc. Am. Math. Soc. 131(7), 2271–2277 (2003)
Yao, G.W.: Convergence of harmonic maps on the Poincaré disk. Proc. Am. Math. Soc. 132(8), 2483–2493 (2004)
Yao, G.W.: Harmonic maps and asymptotic Teichmüller spaces. Manuscr. Math. 122(4), 375–389 (2007)
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The author would like to thank the referee for his careful reading and helpful comments.
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Communicated by Pekka Koskela.
The author was supported by the National Natural Science Foundation of China (Grant No. 11271216).
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Yao, G. On \(C^\infty \)-Compactness of Quasiconformal Harmonic Maps on the Poincaré Disk. Comput. Methods Funct. Theory 16, 365–374 (2016). https://doi.org/10.1007/s40315-015-0148-5
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DOI: https://doi.org/10.1007/s40315-015-0148-5