Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers

Balogh, Zoltán M. ; Leuenberger, Christoph

Canadian journal of mathematics, Tome 51 (1999) no. 5, pp. 915-935

Résumé

Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in ${{\text{C}}^{\text{2}}}$ is studied in details.

Détail

DOI : 10.4153/CJM-1999-040-3

Mots-clés : 32E20, 30C65

Bibliographie

[1] [1] Adams, R. A., Sobolev Spaces. Academic Press, 1975. Google Scholar

[2] [2] Astala, K., Area distortion of quasiconformal mappings. Acta. Math. 173(1994), 37–60. Google Scholar

[3] [3] Astala, K., Planar quasiconformal mappings; deformations and interactions. Preprint. Google Scholar

[4] [4] Astala, K., Balogh, Z. and Reimann, M., Lempert Mappings and Holomorphic Motions in Cn. Astérisque (special volume in honor of A. Douady), to appear. Google Scholar

[5] [5] Balogh, Z., Equivariant Contactomorphisms of Circular Surfaces. Houston J. Math., to appear. Google Scholar

[6] [6] Balogh, Z. and Leuenberger, Chr., Higher Dimensional Riemann Maps. Internat. J. Math., to appear. Google Scholar

[7] [7] Bers, L. and Royden, H. L., Holomorphic families of injections. Acta Math. 157(1986), 259–286. Google Scholar

[8] [8] Bland, J., Contact Geometry and CR structures on S3. Acta Math. 172(1994), 1–49. Google Scholar

[9] [9] Bland, J. and Duchamp, T., Moduli for pointed convex domains. Invent.Math. 104(1991), 61–112. Google Scholar

[10] [10] Černe, M., Stationary Discs of Fibrations Over the Circle. Internat. J. Math. 6(1995), 805–823. Google Scholar

[11] [11] Čirca, E. M., Regularity of boundaries of analytic sets. Math. USSR Sb. 45(1983), 291–336. Google Scholar

[12] [12] ForstneriČ, F., Polynomial hulls of sets fibered over the circle. Indiana Univ.Math. J. 37(1988), 869–889. Google Scholar

[13] [13] Garnett, J. B., Bounded Analytic Functions. Academic Press, 1981. Google Scholar

[14] [14] Globevnik, J., Perturbing analytic discs attached to maximal real submanifolds of Cn. Math. Z. 217(1994), 287–316. Google Scholar

[15] [15] Korányi, A. and Reimann, H. M., Quasiconformal mappings on CR manifolds. Lecture Notes in Math. 1422(1990), 59–75. Google Scholar

[16] [16] Korányi, A. and Reimann, H. M., Foundations for the theory for quasiconformal mappings on the Heisenberg group. Adv. Math. 111(1995), 1–87. Google Scholar

[17] [17] Lempert, L., La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France 109(1981), 427–474. Google Scholar

[18] [18] Lempert, L., A Precise Result on the Boundary Regularity of Biholomorphic Mappings. Math. Z. 193(1986), 559–579. Google Scholar

[19] [19] Lempert, L., Holomorphic invariants, normal forms, and the moduli space of convex domains. Ann. of Math. 128 (1988), 43–78. Google Scholar

[20] [20] Lempert, L., Erratum: A Precise Result on the Boundary Regularity of Biholomorphic Mappings. Math. Z. 206(1991), 501–504. Google Scholar

[21] [21] Lempert, L., On three-dimensional Cauchy-Riemann manifolds. J. Amer.Math. Soc. 5(1992), 923–969. Google Scholar

[22] [22] Mane´, R., Sad, P. and Sullivan, D., On the dynamics of rational maps. Ann. Sci. École Norm. Sup. 16(1983), 193–217. Google Scholar

[23] [23] Reimann, H. M., Quasiconformal mappings and pseudoconvex domains. Proceedings of the XVIth Rolf Nevanlinna Colloquium, Walter de Gruyter & Co., Berlin-New York, 1996, 191–208. Google Scholar

[24] [24] Semmes, S., A Generalization of Riemann Mappings and Geometric Structures on a Space of Domains in Cn. Mem. Amer. Math. Soc. (472) 98, 1992. Google Scholar

[25] [25] Slodkowski, Z., Polynomial hulls with covex fibers and complex geodesics. J. Funct. Anal. 94(1990), 155–176. Google Scholar

[26] [26] Slodkowski, Z., Holomorphic motions and polynomial hulls. Proc. Amer.Math. Soc. 111(1991), 347–357. Google Scholar

[27] [27] Slodkowski, Z., Extension of holomorphic motions. Ann. Scuola Norm. Sup. Pisa 22(1995), 185–210. Google Scholar

[28] [28] Slodkowski, Z., Canonical model for a class of polynomially convex hulls. Math. Ann. 308(1997), 47–63. Google Scholar

[29] [29] Slodkowski, Z., Personal communication. Google Scholar

[30] [30] Sullivan, D. and Thurston, W., Extending holomorphic motions. Acta. Math. 157(1986) 243–257. Google Scholar

[31] [31] Webster, S. M., On the reflection principle in several complex variables. Proc. Amer. Math. Soc. 71(1978), 26–29. Google Scholar