×

zbMATH — the first resource for mathematics

Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. (English) Zbl 0807.32008
Vorliegende Arbeit studiert eine “Randversion” des Schwarz’schen Lemmas. Für jede holomorphe Abbildung \(f:B_ n \to B_ n\) \((B_ n\) bezeichne die Einheitskugel im \(\mathbb{C}^ n)\), die nahe eines Randpunktes von vierter Ordnung mit der Identität von \(B_ n\) übereinstimmt, gilt: \(f \equiv \text{id}_{B_ n}\). Unter Benutzung des Einbettungssatzes von Fornaess und Ergebnissen von Lempert wird obige Aussage auch für streng pseudokonvexe Gebiete bewiesen.
Bekannt ist [vgl. H. Alexander, Math. Ann. 209, 249-256 (1974; Zbl 0281.32019)], daß jede biholomorphe Abbildung \(\varphi:U \cap B_ n \to U' \cap B_ n (n \geq 2)\), die sich zu einer \(C^ 2\)-Abbildung auf den Rand fortsetzen läßt, Restriktion eines Automorphismus von \(B_ n\) ist. Obige Resultate werden benutzt, um Alexander’s Resultate wie folgt zu verschärfen: Sei \(\varphi : B_ n \to B_ n (n \geq 2)\) holomorph, und sei vorausgesetzt, daß \(\varphi\) sich nahe \(P \in \partial B_ n\) und \(\varphi (P) \in \partial B_ n\) als \(C^ 6\)- Abbildung auf den Rand fortsetzen läßt. Schmiegt sich dann \(\varphi (\partial B_ n)\) in \(\varphi (P)\) “hinreichend stark” an \(\partial B_ n\), so ist \(\varphi\) nahe \(P\) “fast” eine biholomorphe Abbildung von \(B_ n\). Genauer: \(\varphi (z) \equiv \psi (z) + 0 (| z - P |^ 3)\) mit \(\psi \in \operatorname{Aut} B_ n\).
Reviewer: P.Pflug (Vechta)

MSC:
32A40 Boundary behavior of holomorphic functions of several complex variables
32T99 Pseudoconvex domains
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
PDF BibTeX Cite
Full Text: DOI
References:
[1] Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. · Zbl 0272.30012
[2] H. Alexander, Holomorphic mappings from the ball and polydisc, Math. Ann. 209 (1974), 249 – 256. · Zbl 0272.32006
[3] L. A. Aĭzenberg, Multidimensional analogues of the Carleman formula with integration over boundary sets of maximal dimension, Multidimensional complex analysis (Russian), Akad. Nauk SSSR Sibirsk. Otdel., Inst. Fiz., Krasnoyarsk, 1985, pp. 12 – 22, 272 (Russian).
[4] Elisabetta Barletta and Eric Bedford, Existence of proper mappings from domains in \?², Indiana Univ. Math. J. 39 (1990), no. 2, 315 – 338. · Zbl 0707.32005
[5] D. Burns, A multi-valued Hartogs theorem and developing maps, preprint.
[6] D. Burns Jr., S. Shnider, and R. O. Wells Jr., Deformations of strictly pseudoconvex domains, Invent. Math. 46 (1978), no. 3, 237 – 253. · Zbl 0412.32022
[7] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219 – 271. · Zbl 0302.32015
[8] Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1 – 65. · Zbl 0289.32012
[9] John Erik Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math. 98 (1976), no. 2, 529 – 569. · Zbl 0334.32020
[10] E. Gavosto, Thesis, Washington University, 1990.
[11] Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in \?\(^{n}\) with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219 – 240. · Zbl 0305.32011
[12] R. E. Greene and Steven G. Krantz, Stability properties of the Bergman kernel and curvature properties of bounded domains, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 179 – 198. · Zbl 0483.32014
[13] Robert E. Greene and Steven G. Krantz, Deformation of complex structures, estimates for the \partial equation, and stability of the Bergman kernel, Adv. in Math. 43 (1982), no. 1, 1 – 86. · Zbl 0504.32016
[14] -, Methods for studying the automorphism groups of weakly pseudoconvex domains, Proc. Internat. Conf. on Complex Geometry (Cetraro, Italy), Mediterranean Press, Calabria, 1991.
[15] Xiao Jun Huang, Some applications of Bell’s theorem to weakly pseudoconvex domains, Pacific J. Math. 158 (1993), no. 2, 305 – 315. · Zbl 0807.32016
[16] -, A boundary rigidity problem of holomorphic mappings on weakly pseudoconvex domains, preprint.
[17] -, Preservation principle of extremal mappings and its applications, Illinois J. Math. (to appear).
[18] Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition, Dover Publications, Inc., New York, 1976. · Zbl 0352.43001
[19] Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. · Zbl 0247.32015
[20] Steven G. Krantz, Function theory of several complex variables, John Wiley & Sons, Inc., New York, 1982. Pure and Applied Mathematics; A Wiley-Interscience Publication. · Zbl 0471.32008
[21] -, A new compactness principle in complex analysis, Univ. Autonoma de Madrid, 1987.
[22] J. J. Kohn and Hugo Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451 – 472. · Zbl 0166.33802
[23] A. G. Vitushkin, V. V. Ezhov, and N. G. Kruzhilin, Extension of local mappings of pseudoconvex surfaces, Dokl. Akad. Nauk SSSR 270 (1983), no. 2, 271 – 274 (Russian). · Zbl 0558.32003
[24] László Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), no. 4, 427 – 474 (French, with English summary). · Zbl 0492.32025
[25] L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8 (1982), no. 4, 257 – 261 (English, with Russian summary). · Zbl 0509.32015
[26] -, Intrinsic distances and holomorphic retracts, Complex Analysis and Applications, Sofia, 1984. · Zbl 0583.32060
[27] S. I. Pinčuk, The analytic continuation of holomorphic mappings, Mat. Sb. (N.S.) 98(140) (1975), no. 3(11), 416 – 435, 495 – 496 (Russian).
[28] Walter Rudin, Holomorphic maps that extend to automorphisms of a ball, Proc. Amer. Math. Soc. 81 (1981), no. 3, 429 – 432. · Zbl 0497.32011
[29] -, Function theory of the unit ball in \( {\mathbb{C}^n}\), Springer-Verlag, Berlin and New York, 1980.
[30] N. Sibony, Unpublished notes.
[31] J. Velling, Thesis, Stanford, 1985.
[32] A. G. Vitushkin, Real-analytic hypersurfaces of complex manifolds, Uspekhi Mat. Nauk 40 (1985), no. 2(242), 3 – 31, 237 (Russian). · Zbl 0588.32025
[33] Warren R. Wogen, Composition operators acting on spaces of holomorphic functions on domains in \?\(^{n}\), Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 361 – 366. · Zbl 0738.47025
[34] H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193 – 233. · Zbl 0158.33301
[35] Shing Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), no. 1, 197 – 203. · Zbl 0424.53040
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.