Abstract
We consider an arbitrary real analytic family Xz,\(z \in \bar D\), over the closed unit disc\(\bar D\), of real analytic plane Jordan curves Xz. Ifj e iθ,e iθ ∋ ∂D, is an arbitrary real-analytic family of orientation-reversing homeomorphisms of\(\bar {\mathbb{C}}\) fixingX e iθ pointwise, we show that there is a unique holomorphic motion of\(\bar {\mathbb{C}}\) extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of X0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlfors, L.V. and Bers, L. Riemann’s mapping theorem for variable metrics,Annals Math.,72(2), 385–404, (1960).
Alexander, H. Continuing 1-dimensional analytic sets,Math. Ann.,191, 143–144, (1971).
Bers, L. and Royden, H.L. Holomorphic families of injections,Acta Math.,157, 259–286, (1986).
Cirka, E.M. Regularity of boundaries of analytic sets,Math USSR-Sb,117, 291–334, (1982), (Russian),Math USSR-Sb,45, 291–336, (1983).
Douady, A. and Earle, C.T. Conformally natural extensions of homeomorphisms of the circle,Acta Math.,157, 23–48, (1986).
Earle, C.T., Kra, I., and Krushkal, S.L. Holomorphic motions and Teichmüller spaces, to appear.
Earle, C.T. and Nag, S. Conformally natural reflections in Jordan curves with applications to Teichmüller spaces, inHolomorphic Functions and Moduli, II, Springer Verlag, New York, 1988, 179–194.
O’Farrell, A.G. and Preskenis, K.J. Approximation by polynomials in two diffeomorphisms,Bull. Am. Math. Soc.,10, 105–107, (1984).
Samelin, T.W., Garnett, T.B., Rubel, L.A., and Shields, A.L. On badly approximable functions,J. Approx. Theory,17, 280–296, (1976).
Garnett, J.B.Bounded Analytic Functions, Academic Press, San Diego, 1981.
Mañe, R., Sad, P., and Sullivan, D. On the dynamics of rational maps,Ann. Sci. Ecole Norm. Sup.,16, 193–217, (1983).
Nikolski, N.K.Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986.
Prössdorf, S.Some Classes of Singular Equations, North-Holland, Amsterdam, 1978.
Slodkowski, Z. Polynomially convex hulls with convex sections and interpolation spaces,Proc. Amer. Math. Soc.,96, 255–260, (1986).
Slodkowski, Z. Holomorphic motions and polynomial hulls,Proc. Amer. Math. Soc.,111, 347–355, (1991).
Slodkowski, Z. Extensions of holomorphic motions,Annali Scuda Norm. Sup. Sev. IV,XXII, 185–210 (1995).
Slodkowski, Z. Holomorphic motions commuting with semigroups,Studia Math.,119, 1–16 (1996).
Sullivan, D. Quasiconformal homeomorphisms and dynamics, III: Topological conjugacy classes of analytic endomorphisms, preprint, 1985.
Sullivan, D. and Thurston, W.P. Extending holomorphic motions,Acta Math.,157, 243–257, (1986).
Wermer, J.Banach Algebras and Several Complex Variables, Markham Publishing Company, Chicago, IL, 1971.
Slodkowski, Z. Orientation-reversing diffeomorphisms and holomorphic motions, preprint.
Slodkowski, Z. Polynomially convex hulls in ℂ3 with totally real fibers,Complex Variables,32, 321–330, (1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by John Mather
Rights and permissions
About this article
Cite this article
Slodkowski, Z. Natural extensions of holomorphic motions. J Geom Anal 7, 637–651 (1997). https://doi.org/10.1007/BF02921638
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02921638