×

zbMATH — the first resource for mathematics

On the quasi-isometries of harmonic quasiconformal mappings. (English) Zbl 1116.30010
The authors study mappings which are harmonic with respect to the Euclidean or Poincaré metric and, in addition, quasiconformal. In both cases they prove a variant of the Ahlfors-Schwarz lemma. For instance they prove the following result. Let \(f: U \to U\) be a \(k\)-quasiconformal Euclidean harmonic mapping onto the unit disk \(U\) in \(R^2.\) Then for all \(z_1,z_2 \in U\) \[ d_h(f(z_1),f(z_2))\leq \frac{1+k}{1-k} d_h(z_1,z_2) \] where \(d_h\) is the hyperbolic metric of \(U.\)

MSC:
30C62 Quasiconformal mappings in the complex plane
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahlfors, L., Conformal invariants, (1973), McGraw-Hill Book Company
[2] Axler, S.; Bourdon, P.; Ramey, W., Harmonic function theory, (1992), Springer Verlag New York · Zbl 0765.31001
[3] Jost, J., Two-dimensional geometric variational problems, (1991), John Wiley & Sons · Zbl 0729.49001
[4] D. Kalaj, Harmonic and quasiconformal functions between convex domains, Doctoral thesis, Beograd, 2002 · Zbl 1220.30032
[5] Lehto, O.; Virtanen, K.I., Quasiconformal mappings in the plane, (1973), Springer-Verlag · Zbl 0267.30016
[6] Mateljević, M., Note on Schwarz lemma, curvature and distance, Zb. radova PMF, 13, 25-29, (1992)
[7] Mateljević, M., Ahlfors – schwarz lemma and curvature, Kragujevac J. math. (zb. radova PMF), 25, 155-164, (2003) · Zbl 1053.30012
[8] Mateljević, M., Estimates for the modulus of the derivatives of harmonic univalent mappings, Rev. roumaine math. pures appl., 47, 5-6, 709-711, (2002) · Zbl 1092.30039
[9] M. Mateljević, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, Rev. Roumaine Math. Pures Appl. 51 (2006)
[10] Schoen, R.; Yau, S.T., Lectures on harmonic maps, () · Zbl 0886.53004
[11] Wan, T., Constant Mean curvature surface, harmonic maps, and universal Teichmüller space, J. differential geom., 35, 643-657, (1992) · Zbl 0808.53056
[12] Yau, S.T., A general Schwarz lemma for kahler manifolds, Amer. J. math., 100, 197-203, (1978) · Zbl 0424.53040
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.