Quasiconformal extension of biholomorphic mappings in several complex variables
Tóm tắt
Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC
n. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of
$$\bar B$$
. We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of
$$\overline {R^{2n} } $$
onto itself and give some applications.
Tài liệu tham khảo
[Be1] J. Becker,Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math.255 (1972), 23–43.
[Be2] J. Becker,Conformal mappings with quasiconformal extensions, inAspects of Contemporary Mathematics (D. Brannan and J. Clunie, eds.), Academic Press, London-New York, 1980, pp. 37–77.
[BePo] J. Becker and C. Pommerenke,Über die quasikonforme Fortsetzung schlichter Funktionen, Math. Z.161 (1978), 69–80.
[Br] A. A. Brodskii,Quasiconformal extension of biholomorphic mappings inTheory of Mappings and Approximation of Functions, Naukova Dunka, Kiew, 1983, pp. 30–34.
[FaKrZy] M. Fait, J. Krzyż and J. Zygmunt,Explicit quasiconformal extensions for some classes of univalent functions, Comment. Math. Helv.51 (1976), 279–285.
[GrHaKo] I. Graham, H. Hamada and G. Kohr,Parametric representation of univalent mappings in several complex variables, Canad. J. Math.54 (2002), 324–351.
[GrKo] I. Graham and G. Kohr,Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, New York, 2003.
[HaKo] H. Hamada and G. Kohr,Loewner chains and quasiconformal extension of holomorphic mappings, Ann. Polon. Math.81 (2003), 85–100.
[Kr] J. Krzyż,Convolution and quasiconformal extension, Comment. Math. Helv.51 (1976), 99–104.
[Pf1] J. A. Pfaltzgraff,Subordination chains and univalence of holomorphic mappings in C n, Math. Ann.210 (1974), 55–68.
[Pf2] J. A. Pfaltzgraff,Subordination chains and quasiconformal extension of holomorphic maps in C n, Ann. Acad. Sci. Fenn. Ser. A I Math.1 (1975), 13–25.
[ReMa] F. Ren and J. Ma,Quasiconformal extension of biholomorphic mappings of several complex variables, J. Fudan Univ. Nat. Sci.34 (1995), 545–556.
[RoSu] K. Roper and T. J. Suffridge,Convexity properties of holomorphic mappings in C n, Trans. Amer. Math. Soc.351 (1999), 1803–1833.
[Sa] S. Saks,Theory of the Integral, English translation by L. C. Young; with two additional notes by Stefan Banach.—2nd rev. ed., G. E. Stechert, New York, 1937.
[Su] T. J. Suffridge,Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, inComplex Analysis, Lecture Notes in Math.599, Springer-Verlag, Berlin, 1976, pp. 146–159.
[Vä] J. Väisälä,Lectures on n-dimensional Quasiconformal Mappings, Lecture Notes in Math.229, Springer-Verlag, Berlin-New York, 1971.