×

Loewner chains and parametric representation in several complex variables. (English) Zbl 1029.32004

Let \(B\) be the unit ball of \(\mathbb{C}^n\) with respect to an arbitrary norm. The authors study certain properties of Loewner chains and their transition mappings on the unit ball \(B\). They show that any Loewner chain \(f(z,t)\) and the transition mapping \(v(z,s,t)\) associated to \(f(z,t)\) satisfy locally Lipschitz conditions in \(t\) locally uniformly with respect to \(z\in B\). Moreover, they prove that a mapping \(f\in H(B)\) has parametric representation if and only if there exists a Loewner chain \(f(z,t)\) such that the family \(\{e^{-t}f(z,t)\}_{t\geq 0}\) is a normal family on \(B\) and \(f(z)=f(z,0)\). The authors also show that univalent solutions \(f(z,t)\) of the generalized Loewner differential equation in higher dimensions are unique when \(\{e^{-t}f(z,t)\}_{t\geq 0}\) is a normal family in \(B\). Finally, they show that the set of mappings which have parametric representation on \(B\) is compact.

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32A17 Special families of functions of several complex variables
32A19 Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Becker, J., Über die Lösungsstruktur einer differentialgleichung in der konformen abbildung, J. reine angew. math., 285, 66-74, (1976) · Zbl 0324.30035
[2] Becker, J., Conformal mappings with quasiconformal extensions, (), 37-77 · Zbl 0491.30012
[3] Brodskii, A.A., Quasiconformal extension of biholomorphic mappings, (), 30-34
[4] Chuaqui, M., Applications of subordination chains to starlike mappings in \(C\^{}\{n\}\), Pacific J. math., 168, 33-48, (1995) · Zbl 0822.32001
[5] Graham, I.; Hamada, H.; Kohr, G., Parametric representation of univalent mappings in several complex variables, Canad. J. math., 54, 324-351, (2002) · Zbl 1004.32007
[6] Graham, I.; Hamada, H.; Kohr, G.; Suffridge, T.J., Extension operators for locally univalent mappings, Michigan math. J., 50, 37-55, (2002) · Zbl 1025.32017
[7] Graham, I.; Kohr, G., An extension theorem and subclasses of univalent mappings in several complex variables, Complex variables, 47, 59-72, (2002) · Zbl 1026.32033
[8] I. Graham, G. Kohr, Geometric Function Theory in One and Higher Dimensions, Dekker, New York, to appear · Zbl 1042.30001
[9] Graham, I.; Kohr, G.; Kohr, M., Loewner chains and the roper – suffridge extension operator, J. math. anal. appl., 247, 448-465, (2000) · Zbl 0965.32008
[10] Hamada, H., Starlike mappings on bounded balanced domains with C1-plurisubharmonic defining functions, Pacific J. math., 194, 359-371, (2000) · Zbl 1018.32018
[11] H. Hamada, Univalence and quasiconformal extension of holomorphic maps on balanced pseudoconvex domains, preprint
[12] Hamada, H.; Kohr, G., Subordination chains and the growth theorem of spirallike mappings, Mathematica (cluj), 42, 153-161, (2000) · Zbl 1027.46094
[13] Hamada, H.; Kohr, G., Subordination chains and univalence of holomorphic mappings on bounded balanced pseudoconvex domains, Ann. univ. mariae Curie-skłodowska sect. A, 55, 61-80, (2001) · Zbl 1018.32019
[14] Hamada, H.; Kohr, G., The growth theorem and quasiconformal extension of strongly spirallike mappings of type α, Complex variables, 44, 281-297, (2001) · Zbl 1026.32035
[15] H. Hamada, G. Kohr, Loewner chains and quasiconformal extension of holomorphic mappings, Ann. Polon. Math., to appear · Zbl 1030.30020
[16] Kohr, G., Using the method of Löwner chains to introduce some subclasses of biholomorphic mappings in \(C\^{}\{n\}\), Rev. roumaine math. pures appl., 46, 743-760, (2001) · Zbl 1036.32014
[17] Kohr, G.; Liczberski, P., Univalent mappings of several complex variables, (1998), Cluj Univ. Press Cluj-Napoca
[18] Kubicka, E.; Poreda, T., On the parametric representation of starlike maps of the unit ball in \(C\^{}\{n\}\) into \(C\^{}\{n\}\), Demonstratio math., 21, 345-355, (1988) · Zbl 0674.32014
[19] Narasimhan, R., Several complex variables, (1971), Univ. of Chicago Press Chicago · Zbl 0223.32001
[20] Pfaltzgraff, J., Subordination chains and univalence of holomorphic mappings in \(C\^{}\{n\}\), Math. ann., 210, 55-68, (1974) · Zbl 0275.32012
[21] Pfaltzgraff, J.A., Subordination chains and quasiconformal extension of holomorphic maps in \(C\^{}\{n\}\), Ann. acad. sci. fenn. ser. A, 1, 13-25, (1975) · Zbl 0314.32001
[22] Pfaltzgraff, J.A., Loewner theory in \(C\^{}\{n\}\), Amer. math. soc., 11, 46, (1990)
[23] Pfaltzgraff, J.A.; Suffridge, T.J., Close-to-starlike holomorphic functions of several variables, Pacific J. math., 57, 271-279, (1975) · Zbl 0319.32011
[24] Pommerenke, C., Univalent functions, (1975), Vandenhoeck and Ruprecht Göttingen
[25] Poreda, T., On the univalent holomorphic maps of the unit polydisc of \(C\^{}\{n\}\) which have the parametric representation, I—the geometrical properties, Ann. univ. mariae Curie-skłodowska sect. A, 41, 105-113, (1987) · Zbl 0698.32004
[26] Poreda, T., On the univalent subordination chains of holomorphic mappings in Banach spaces, Comment. math., 128, 295-304, (1989) · Zbl 0694.46033
[27] Ren, F.; Ma, J., Quasiconformal extension of biholomorphic mappings of several complex variables, J. fudan univ. nat. sci., 34, 545-556, (1995) · Zbl 0856.32016
[28] Royden, H.L., Real analysis, (1988), Macmillan New York · Zbl 0704.26006
[29] Roper, K.; Suffridge, T.J., Convexity properties of holomorphic mappings in \(C\^{}\{n\}\), Trans. amer. math. soc., 351, 1803-1833, (1999) · Zbl 0926.32012
[30] Suffridge, T.J., Starlike and convex maps in Banach spaces, Pacific J. math., 46, 474-489, (1973) · Zbl 0263.30016
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.