Journal of Zhejiang University SCIENCE A
(Monthly)

2006   Vol. 7   Suppl. II   p. 198-202

A problem on extremal quasiconformal extensions

Chen Zhi-Guo

(Department of Mathematics, Zhejiang University, Hangzhou 310027, China)
E-mail: zgchen@zju.edu.cn
Received Nov. 20, 2005 revision accepted Mar. 21, 2006

Abstract: In this paper we give a short survey on a problem on extremal quasiconformal extensions. It had been a conjecture for a long time that the dilatations K₀(h) and K₁(h) are equal before Anderson and Hinkkanen disproved this by constructing concrete examples of a family of affine mappings of some parallelograms. The problem also engendered many interesting results. At the end of the current paper, we discuss relationships among K₀(h), H(h) and K₁(h) as a concluding remark.

Key words: Quasisymmetric mapping, Extremal quasiconformal mapping, Universal Teichmüller space, Non-Strebel point
doi:10.1631/jzus.2006.AS0198             CLC number: O174.5

References:

[1] Ahlfors, L.V., 1966. Lectures on Quasiconformal Mappings. Van Nostrand, New York.

[2] Anderson, J.M., Hinkkanen, A., 1995. Quadrilaterals and extremal quasiconformal extensions. Comment. Math. Helv., 70:455-474.

[3] Chen, J., Chen, Z., 1997. A Remark on “An approximation condition and extremal quasiconformal extensions”. Chinese Science Bull., 42:1765-1767.

[4] Chen, J., Zhu, H., Liang, X., 2002. Quadrilateral and substantial boundary points. Chinese Annals of Math., 23A(2):197-204 (in Chinese).

[5] Earle, C., Li, Z., 1999. Isometrically embedded polydisks in infinite dimensional Teichmüller spaces. J. Geom. Anal., 9(1):51-71.

[6] Fehlmann, R., 1982. Quasiconformal mappings with free boundary components. Annal. Acad. Sci. Fenn., A.I. Math., 7:337-347.

[7] Krushkal, S.L., 2003. Polygon quasiconformal maps and Grunsky inequalities. J. Anal. Math., 90:175-196.

[8] Kühnau, R.K., 2000. Drei Funktionale eines Quasikreises. Annal. Acad. Sci. Fenn., A.I. Math., 25:413-415.

[9] Lakic, N., 1995. Strebel Points, Lipa’s Legacy. Proceedings of the Bers Colloquium, New York, p.417-431.

[10] Lehto, O., Virtanen, K.I., 1973. Quasiconformal Mappings in the Plane. Springer, Berlin.

[11] Li, Z., Wu, S., Qi, Y., 1999. A class of quasisymmetric mappings and substantial boundary points. Chinese Science Bull., 44:926-930 (in Chinese).

[12] Liang, X., Zhu, H., 2001. An extremal problem of quasi-conformal mappings in hyperbolic regions. Journal of Fudan University, 40(6):645-648.

[13] Qi, Y., 1998. A problem in extremal quasiconformal extensions. Sci. in China, Series A, 41(11):1135-1141.

[14] Reich, E., 1997. An approximation condition and extremal quasiconformal extensions. Proc. Amer. Math. Soc., 125(5):1479-1481.

[15] Reich, E., Strebel, K., 1974. Extremal Quasiconformal Mappings with Given Boundary Values. In: Contributions to Analysis. Academic Press, p.375-391.

[16] Shen, Y., 2000. A counterexample theorem in extremal quasiconformal mapping theory. Sci. in China, Series A, 43(9):929-936.

[17] Strebel, K., 1976. On the existence of extremal Teichmüller mappings. J. D’Analyse Math., 30:464-480.

[18] Strebel, K., 1999. On the dilatation of extremal quasiconformal mappings of polygons. Comment. Math. Helv., 74(1):143-149.

[19] Wu, S., 1997. Moduli of quadrilaterals and extremal quasi-conformal extensions of quasisymmetric functions. Comment. Math. Helv., 72(4):593-604.

[20] Yang, S., 1997. On dilatations and substantial boundary points of homeomorphims of Jordan curves. Results in Math., 31:180-188.