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A Boundary Rigidity Problem For Holomorphic Mappings on Some Weakly Pseudoconvex Domains

Published online by Cambridge University Press:  20 November 2018

Xiaojun Huang
Xiaojun Huang*
Affiliation:
Department of Mathematics Washington University St. Louis, Missouri 63130 U.S.A e–mail: c31995xh@wuvmd.wustl.edu
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Abstract

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In this paper, we study the boundary version of the classical Cartan theorem. We show that for some weakly pseudoconvex domains, when a holomorphic self-mapping has a sufficiently high order of contact (which depends only on the geometric properties of the domains) with the identical map at some boundary point, then it must coincide with the identity.

Keywords

32H0232H15Burns-Krantz rigidity problemsiteration of self-mappingsKobayashi metrics and Kobayashi distancescomplex geodesiesholomorphic retractshorospherespseudoconvex domains of finite type.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 405 - 420
Copyright
Copyright © Canadian Mathematical Society 1995

References

[Abl] Abate, M., Boundary behavior of invariant distances and complex geodesies, Atti Accad. Naz. Lincei Rend. 80(1986), 100–106.Google Scholar
[Ab2] Abate, M., Horospheres and iterates of holomorphic maps, Math. Z. 198(1988), 225–238.Google Scholar
(Ab3J Abate, M., Iteration Theory of Holomorphic Maps on Taut Manifold, Mediterranean Press, Rendre, Cosenza, 1989.Google Scholar
[B] Bedford, E., On the automorphism group of a Stein manifold, Math. Ann. 266(1983), 215–227.Google Scholar
[Bpl] Bedford, E. and Pinchuk, S., Domains in C2 with noncompact group of automorphisms, Math. USSR-Sb. 63(1989), 141–151.Google Scholar
[BP2] Bedford, E., Holomorphic tangent vector fields: Domains with noncompact automorphism groups, preprint.Google Scholar
[Bel] Bell, S., Mapping problems in complex analysis and the d-problems, Bull. Amer. Math. Soc. (N.S.) (2) 22(1990), 233–259.Google Scholar
[Be2] Bell, S., Compactness of families of holomorphic mappings up to the boundary, Lecture Notes in Math. 1268, Springer, 29–42.Google Scholar
[BD] Bell, S. and Catlin, D., Boundary regularity of proper holomorphic mappings, Duke Math. J. 49(1982), 385–396.Google Scholar
[BK] Burns, D., Jr. and Krantz, S., A new rigidity property for holomorphic mappings, J. Amer. Math. Soc., to appear.Google Scholar
[Ca] Catlin, D., Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), 429–266.Google Scholar
[CHL] Chang, C.H., Hu, M.C. and Lee, H.P., Extremal discs with boundary data, Trans. Amer. Math. Soc. 310(1988), 355–369.Google Scholar
[Co] Cho, S., A lower bound on the Kobayashi metric near a point of finite type in Cn, preprint.Google Scholar
[D] Dapos;Angelo, J.P. , Real hypersurfaces, orders of contact, and applications, Ann. of Math. 115(1982),615—637.Google Scholar
[DF1] Diederich, K. and Fornaess, J.E., Pseudoconvex domains; Bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39(1977), 129–141.Google Scholar
[DF2] Diederich, K., Boundary regularity of proper holomorphic mappings, Invent. Math. 67(1982), 363–384.Google Scholar
[DF3] Diederich, K., Proper holomorphic maps onto pseudoconvex domains with real analytic boundaries, Ann. of Math. 107(1979), 575–592.Google Scholar
[G] Graham, I., Boundary behavior of the Carathedory and Kobayashi metrics on strongly pseudoconvex domains in Cn with smooth boundary, Trans. Amer. Math. Soc. 207(1975), 219–240.Google Scholar
[HI] Huang, X., Some applications of Bell's theorem on weakly pseudoconvex domains, Pacific J. Math. 158 (1993), 305–315.Google Scholar
[H2] Huang, X., A preservation principle of Extremal mappings near a strongly pseudoconvex point and its applications, Illinois J. Math. (1) 38(1994), to appear.Google Scholar
[Kl] Krantz, S., A new compactness principle in complex analysis, Univ. Antonoma de Madrid, 1986.Google Scholar
[K2] Krantz, S., Function Theory of Several Complex Variables (second edition), Wadsworth, 1992.Google Scholar
[LI] Lempert, L., La métrique de Kobayashi et la representation des domaines sur la boule, Bull. Soc. Math. France. 109(1981), 427–474.Google Scholar
[L2] Lempert, L., Intrinsic distances and holomorphic retracts. In: Complex analysis and applications ‘81, Varns, Bulgarian Academy of Sciences, Sofia, 1984. 341–364.Google Scholar
[L3] Lempert, L., Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8(1982), 257–261.Google Scholar
[M] Mercer, Peter, Complex geodesies and iterates of holomorphic maps on convex domains in Cn, Trans. Amer. Math. Soc. 338(1993), 201–211.Google Scholar
[RW1 Royden, H.L. and Wong, P.M., Carathéodory and Kobayashi metrics on convex domains, preprint, 1983.Google Scholar