Biholomorphic invariants related to the Bergman functions

Skwarczyński, Maciej

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Biholomorphic invariants related to the Bergman functions

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Maciej Skwarczyński

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Rozprawy Matematyczne tom/nr w serii: 173 wydano: 1980

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CONTENTS

PRELIMINARY REMARKS........................................................................................ 5

Introduction..................................................................................................... 5
Basic definitions, examples and facts............................................................... 8

I. LU QI-KENQ DOMAINS........................................................................................... 13

Some properties of Lu Qi-keng domains.................................................. 13
An example of bounded non-Lu Qi-keng domain............................................ 14
Doubly connected Lu Qi-keng domains in the plane...................................... 15

II. REPRESENTATIVE COORDINATES................................................................... 16

The Bergman metric tensor......................................................................... 16
A property of representative coordinates........................................................... 19

III. AN INVARIANT DISTANCE................................................................................... 20

Biholomorphic mappings and canonical isometry................................. 20
Critical points of the invariant distance.............................................................. 22
Completeness with respect to the invariant distance..................................... 22

IV. EXTENSION THEOREM....................................................................................... 27

Semiconformal mappings........................................................................... 27
Extension theorem................................................................................................ 31
Local characterization of a biholomorphic mapping...................................... 33

V. DOMAIN DEPENDENCE...................................................................................... 36

Ramadanov theorem.................................................................................... 36
An analogue of Ramadanov theorem for decreasing sequences................ 37
A counterexample in the plane............................................................................ 39

VI. THE IDEAL BOUNDARY....................................................................................... 40

Definition of the ideal boundary................................................................... 40
Characteristic properties....................................................................................... 49
The case of bounded circular domains.............................................................. 53
Plane domains and strictly pseudoconvex domains........................................ 54

REFERENCES.............................................................................................................. 58

Słowa kluczowe

Tematy

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

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Rozprawy Matematyczne tom/nr w serii: 173

Liczba stron

59

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Opis fizyczny

Dissertationes Mathematicae, Tom CLXXIII

Daty

wydano

1980

Twórcy

autor

Maciej Skwarczyński

Bibliografia

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[3] S. Bergman, Über die Existenz von Repräsentantenbereichen, Math. Ann. 102 (1929), p. 430-446,
[4] S. Bergman, On pseudo-conformal mappings of circular domains, Pacific J. Math. 41 (1972), p. 679-585.
[5] S. Bergman, The kernel function and conformal mapping, Math. Surveys No. 5, Amor. Math. Soc. (1950).
[6] S. Bergman, and. M. Schiffer, Kernel functions and conformal mapping, Compositio Math. 8 (1961), p. 206-249.
[7] H. Bremermann, Holomorphic continuation of the kernel function and the Bergman metric in several complex variables, Lectures on functions of a complex variable, Ann Arbor, Univ. of Michigan Press, 1965.
[8] H. Cartan, Les fonctions de deux variable complexes et le problème de la représentation analytique, J. Math. Pures Appl. 10 (1931), p. 1-114.
[9] B. Chalmers, Sur le comportement, a la frontière de la fonction noyau do Bergman, Comp. Rend. Acad. Sci. 266, 33 (1968).
[10] A. M. Davie and B. K. Oksendal, Peak interpolation sets for some algebras of analytic functions, Pacific J. Math. 41 (1972), p. 81-87.
[11] Ch. Fefermann, The Bergman kernel and biholomorphic mappings of pseudo-convex domains, Invent. Math. 26 (1974), p. 1-65.
[12] B. A. Fuks, Theory of analytic functions of several complex variables, OGIZ, Moscow 1948.
[13] J. Hadamard, Sur le principe de Dirichlet, Bull. Soc. Math. France 34 (1906), p. 136-139.
[14] V. Havin, Approximation in mean by analytic functions, Dold. Akad. Nauk 178 (1968), p. 1025-1029.
[15] L, Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), p. 299-319.
[16] L. Helms, Introduction to potential theory, Wiley Interscience, New York 1969.
[17] L. Hormander, An introduction to complex analysis in several variables, Van Nostrand, 1966.
[18] Hua Lo-Keng, Harmonic analysis of several complex variables in the classical domains, Science Press, Poking 1958; Engl, transl., Transl. Math. Monographs 6, Amer. Math. Soc. Providence, R. I., 1963.
[19] T. B. Iwiński and M. Skwarczyński, The convergence of Bergman functions for a decreasing sequence of domains, Z. Ciesielski and J. Musielak eds., Approximation theory, p. 117-120, P.W.N., Warszawa 1975.
[20] N, Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), p. 149-158.
[21] S. Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1969), p. 267-290.
[22] A. Lichnerowicz, Variétés complexes et tenseur de Bergman, Ann. lust. Fourier 15 (1965), p. 345-408.
[23] Lu Qi-keng, On Kaehler manifolds with constant curvature, Acta Math. Sinica 15 (1966), p. 269-281.
[24] Г. А. Маргулис, Соответствие границ при биголоморфпых отображениях многомерных областей, Тезисы докладов Всесоювной Конференции по теории функций, Харков (1971), р. 137-138,
[25] А. И. Маркушевич, Теория аналитических функций, vol. I and II, Наука, Москва 1967.
[26] S. Matsuura, On the Lu Qi-keng conjecture and the Bergman representative domains, Pacific J. Math. 49 (1973), p. 407-416.
[27] I. Ramadanov, Sur une propriété de la fonction de Bergman, Comt. Rend. Acad. Bulg. Sci. 20 (1967), p. 759-762.
[28] P. L. Rosenthal, On the zeros of the Bergman function in doubly-connected domains, Proc. Amer. Math. Soc. 21 (1969), p. 33-35.
[29] S. Sinanian, Approximation in the mean by analytic functions and polynomials on the complex plane, Mat. Sbor. 69 (1966), p. 111.
[30] M. Skwarczyński, The Bergman function and semiconformal mappings, Thesis, Warsaw University, 1969.
[31] M. Skwarczyński, Bergman function and conformal mappings in several complex variables, Bull. Acad. Polon. Sci.^7 (1969), p. 139.
[32] M. Skwarczyński, The invariant distance in the theory of pseudoconformal transformations and the Lu Qi-keng conjecture, Proc. Amer. Math. Soc. 22 (1969), p. 305-310.
[33] M. Skwarczyński, The Bergman function and semiconformal mappings, Bull. Acad. Polon. Sci. 22 (1974), p. 667-673.
[34] M. Skwarczyński, A new notion of boundary in the theory of several complex variables, ibidem 24 (1976), p. 327-330.
[35] G. Springer, Pseudo-conformal transformations onto circular domains, Duke Math. J. 18 (1951), p. 411-424.
[36] N. Suita and A. Yamada, On the Lu Qi-keng conjecture, Proc. Amer. Math. Soc. 59 (1976), p. 222-224.
[37] N. Vormoor, Topologische Fortsetzung biholomorpher Funktionen auf dem Bande bei beschränkten streng-pseudo-konvexen Gebieten im 4C^n$ mit $C^∞$-Band, Math. Ann. 204 (1973), p. 239-269.
[38] H. Welke, Über die analytischen Abbildungen von Kreiskörpern und Hartog-schen Bereichen, ibidem 103 (1930), p. 437-449
[39] K. Zarankiewicz, Über ein numerisches Verfahren zur konformen Abbildung zweifach zusammenhangender Gebiete, Zeitschr. Angew, Math. Mech. 14 (1934), p. 97-104.
[40] S. Zaremba, Sur le calcul numérique des fonctions demandées dans le problème de Dirichlet et le problème hydrodynamique, Bull. Internat. Acad. Sci. Cracovie 1 (1909), p. 125-195.
[41] Б. Зиновьев, О воспроизводящих ядрах для кратнокруговых областей голоморфности, Сиб. мат. ж. 15 (1974), р. 35-48.

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Biholomorphic invariants related to the Bergman functions

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