zbMATH — the first resource for mathematics

Inner estimate and quasiconformal harmonic maps between smooth domains. (English) Zbl 1173.30311
Summary: We prove a type of “inner estimate” for quasi-conformal diffeomorphisms, which satisfies a certain estimate concerning their Laplacian. This, in turn, implies that quasiconformal harmonic mappings between smooth domains (with respect to an approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz. We discuss harmonic mappings with respect to (a) spherical and Euclidean metrics (which are approximately analytic) (b) the metric induced by a holomorphic quadratic differential.

30C62 Quasiconformal mappings in the complex plane
30C25 Covering theorems in conformal mapping theory
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
Full Text: DOI
[1] L. Ahlfors,Lectures on Quasiconformal Mappings. Van Nostrand, Princeton, N.J., 1966. · Zbl 0138.06002
[2] I. Anić, V. Marković and M. Mateljević,Uniformly bounded maximal {\(\sigma\)}-discs and Bers space and harmonic maps, Proc. Amer. Math. Soc.128 (2000), 2947–2956 · Zbl 0957.30032
[3] A. Beurling and L. Ahlfors,The boundary correspondence under quasiconformal mappings, Acta Math.96 (1956), 125–142. · Zbl 0072.29602
[4] J. Clunie and T. Sheil-Small,Harmonic univalent functions. Ann. Acad. Sci. Fenn. A I Math.9 (1984), 3–25. · Zbl 0506.30007
[5] P. Duren.Harmonic Mappings in the Plane, Cambridge Univ. Press, 2004. · Zbl 1055.31001
[6] Z-C. Han,Remarks on the geometric behavior of harmonic maps between surfaces. Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), A. K. Peters, Wellesley, MA, 1996, pp. 57–66.
[7] G. M. Goluzin,Geometric Function Theory, Nauka, Moscow, 1966 (Russian). · Zbl 0083.06604
[8] E. Heinz,On certain nonlinear elliptic differential equations and univalent mappings, J. Analyse Math.5 (1956/57), 197–272. · Zbl 0085.08701
[9] D. Kalaj,Harmonic Functions and Harmonic Quasiconformal Mappings Between Convex Domains, Thesis, Beograd, 2002. · Zbl 1093.31002
[10] D. Kalaj,On harmonic diffeomorphisms of the unit disc onto a convex domain, Complex Var. Theory Appl.48 (2003), 175–187. · Zbl 1041.30006
[11] D. Kalaj,Quasiconformal harmonic functions between convex domains, Publ. Inst. Math. (Beograd) (N. S.),76(90) (2004), 3–20. · Zbl 1220.30032
[12] D. Kalaj,Quasiconformal harmonic functions between Jordan domains, preprint. · Zbl 1151.30014
[13] D. Kalaj and M. Pavlović,Boundary correspondence under harmonic quasiconformal homeomorphisms of a half-plane. Ann. Acad. Sci. Fenn. Math.30 (2005), 159–165. · Zbl 1071.30016
[14] O. Kellogg,On the derivatives of harmonic functions on the boundary. Trans. Amer. Math. Soc.33 (1931), 689–692. · Zbl 0002.02702
[15] M. Knežević and M. Mateljević,Onq. c. and harmonic mapping and the quasi-isometry, preprint.
[16] P. Li., L. Tam and J. Wang,Harmonic diffeomorphisms between hyperbolic Hadamard manifolds, to appear in J. Geom. Anal. · Zbl 0855.58020
[17] O. Martio,On Harmonic Quasiconformal Mappings, Ann. Acad. Sci. Fenn. Ser. A I 425 (1968). · Zbl 0162.37902
[18] M. Mateljević,Note on Schwarz lemma, curvature and distance, Zb. Rad. (Kragujevac)16 (1994), 47–51. · Zbl 0828.53014
[19] M. Mateljević,Ahlfors-Schwarz lemma and curvature, Kragujevac J. Math.25 (2003), 155–164.
[20] M. Mateljević,Dirichlet’s principle, distortion and related problems for harmonic mappings, Publ. Inst. Math. (Belgrad) (N. S.)75(89), (2004), 147–171. · Zbl 1081.30022
[21] M. Mateljević,Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, submitted to Proceedings of the X-th Romanian-Finnish Seminar, August 14–19, 2005, Cluj-Napoca.
[22] M. Pavlović,Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disc, Ann. Acad. Sci. Fenn.27 (2002), 365–372. · Zbl 1017.30014
[23] C. Pommerenke,Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992. · Zbl 0762.30001
[24] Y. Shi, L. Tam and T. Wan,Harmonic maps on hyperbolic spaces with singular boundary values. J. Diff. Geom.51 (1999), 551–600. · Zbl 1030.58009
[25] L. Tam and T. Wan,Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials, Comm. Anal. Geom.4 (1994), 593–625. · Zbl 0842.58014
[26] L. Tam and T. Wan,Quasiconformal harmonic diffeomorphism and universal Teichmüler space, J. Diff. Geom.42 (1995), 368–410. · Zbl 0873.32019
[27] L. Tam and T. Wan,On quasiconformal harmonic maps, Pacific J. Math.182 (1998), 359–383. · Zbl 0892.58017
[28] T. Wan,Constant mean curvature surface, harmonic maps, and universal Teichmüller space, J. Diff. Geom.35 (1992), 643–657. · Zbl 0808.53056
[29] A. Zygmund,Trigonometric Series I, Cambridge University Press, 1958. · Zbl 0628.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.