Two intrinsic pseudo-metrics with pseudoconvex indicatrices and starlike circular domains

Kazuo AZUKAWA

doi:10.2969/jmsj/03840627

Kazuo AZUKAWA

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JOURNAL FREE ACCESS

1986 Volume 38 Issue 4 Pages 627-647

DOI https://doi.org/10.2969/jmsj/03840627

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Published: 1986 Received: March 29, 1985 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Tadashi Kuroda on his 60th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) K. Azukawa, Hyperbolicity of circular domains, Tohoku Math. J., 35 (1983), 403-413.
2) K. Azukawa, Square-integrable holomorphic functions on a circular domain in Cⁿ, Tohoku Math. J., 37 (1985), 15-26.
3) K. Azukawa and J. Burbea, Hessian quartic forms and the Bergman metric, Kodai Math. J., 7 (1984), 133-152.
4) T. J. Barth, The Kobayashi indicatrix at the center of a circular domain, Proc. Amer. Math. Soc., 88 (1983), 527-530.
5) H. J. Bremerman, Holomorphic Continuation of the Kernel Function and the Bergman Metric in Several Complex Variables, Lectures on Functions of a Complex Variable, Univ. Michigan Press, 1955.
6) J. Burbea, Inequalities between intrinsic metrics, Proc. Amer. Math. Soc., 67 (1977), 50-54.
7) C. Carathéodory, Über die Geometrie der analytischen Abbildungen, die durch analytische Funktionen von zwei Veränderlichen vermittelt werden, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 96-145.
8) H. Cartan, Les fonctions de deux variables complexes et le problème de la représentation analytique, J. Math. Pures Appl. (9), 10 (1931), 1-114.
9) H. Grauert und R. Remmert, Plurisubharmonische Funktionen in komplexen Räume, Math. Z., 65 (1956), 175-194.
10) S. Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc., 92 (1959), 267-290.
11) S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc., 82 (1976), 357-416.
12) S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, New York, 1977.
13) A. Kodama, On boundedness of circular domains, Proc. Japan Acad., 58 (1982), 227-230.
14) J. Mehring und F. Sommer, Kernfunktion und Hüllenbildung in der Funktionentheorie mehrerer Veränderlichen, Math. Ann., 131 (1956), 1-16.
15) P. Pflug, Quadraintegrable holomorphe Funktionen und die Serre-Vermutung, Math. Ann., 216 (1975), 285-288.
16) P. Pflug, About the Carathéodory completeness of all Reinhardt domains, Functional Analysis, Holomorphy and Approximation Theory II, G. I. Zapata (ed.), North-Holland, Amsterdam, 1984, pp. 331-337.
17) H. L. Royden, Remarks on the Kobayashi metric, Lecture Notes in Math., 185, Springer, 1971, pp. 125-137.
18) A. Sadullaev, Schwarz lemma for circular domains and its applications, Math. Notes Aca. Sci. USSR, 27 (1980), 120-125.
19) N. Sibony, A class of hyperbolic manifolds, Ann. of Math. Studies, 100, Princeton Univ. Press, 1981, pp. 357-372.
20) M. Skwarczynski, Biholomorphic invariants related to the Bergman functions, Dissertationes Math., 173 (1980), 1-59.
21) M. Suzuki, The intrinsic metrics on the circular domains in Cⁿ, Pacific J. Math., 112 (1984), 249-256.
22) V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables, M. I. T. Press, Massachusetts, 1966.
23) H. Wu, Normal families of holomorphic mappings, Acta Math., 119 (1967), 193-233.

Right : [1] K. Azukawa, Hyperbolicity of circular domains, Tohoku Math. J., 35 (1983), 403-413.
[2] K. Azukawa, Square-integrable holomorphic functions on a circular domain in Cⁿ, Tôhoku Math. J., 37 (1985), 15-26.
[3] K. Azukawa and J. Burbea, Hessian quartic forms and the Bergman metric, Kodai Math. J., 7 (1984), 133-152.
[4] T. J. Barth, The Kobayashi indicatrix at the center of a circular domain, Proc. Amer. Math. Soc., 88 (1983), 527-530.
[5] H. J. Bremerman, Holomorphic Continuation of the Kernel Function and the Bergman Metric in Several Complex Variables, Lectures on Functions of a Complex Variable, Univ. Michigan Press, 1955.
[6] J. Burbea, Inequalities between intrinsic metrics, Proc. Amer. Math. Soc., 67 (1977), 50-54.
[7] C. Carathéodory, Über die Geometrie der analytischen Abbildungen, die durch analytische Funktionen von zwei Veränderlichen vermittelt werden, Abh. Math. Sem. Univ. Hamburg, 6 (1928), 96-145.
[8] H. Cartan, Les fonctions de deux variables complexes et le problème de la représentation analytique, J. Math. Pures Appl. (9), 10 (1931), 1-114.
[9] H. Grauert und R. Remmert, Plurisubharmonische Funktionen in komplexen Räume, Math. Z., 65 (1956), 175-194.
[10] S. Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc., 92 (1959), 267-290.
[11] S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc., 82 (1976), 357-416.
[12] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, New York, 1977.
[13] A. Kodama, On boundedness of circular domains, Proc. Japan Acad., 58 (1982), 227-230.
[14] J. Mehring und F. Sommer, Kernfunktion und Hüllenbildung in der Funktionentheorie mehrerer Veränderlichen, Math. Ann., 131 (1956), 1-16.
[15] P. Pflug, Quadraintegrable holomorphe Funktionen und die Serre-Vermutung, Math. Ann., 216 (1975), 285-288.
[16] P. Pflug, About the Carathéodory completeness of all Reinhardt domains, Functional Analysis, Holomorphy and Approximation Theory II, G. I. Zapata (ed.), North-Holland, Amsterdam, 1984, pp. 331-337.
[17] H. L. Royden, Remarks on the Kobayashi metric, Lecture Notes in Math., 185, Springer, 1971, pp. 125-137.
[18] A. Sadullaev, Schwarz lemma for circular domains and its applications, Math. Notes Aca. Sci. USSR, 27 (1980), 120-125.
[19] N. Sibony, A class of hyperbolic manifolds, Ann. of Math. Studies, 100, Princeton Univ. Press, 1981, pp. 357-372.
[20] M. Skwarczynski, Biholomorphic invariants related to the Bergman functions, Dissertationes Math., 173 (1980), 1-59.
[21] M. Suzuki, The intrinsic metrics on the circular domains in Cⁿ, Pacific J. Math., 112 (1984), 249-256.
[22] V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables, M. I. T. Press, Massachusetts, 1966.
[23] H. Wu, Normal families of holomorphic mappings, Acta Math., 119 (1967), 193-233.

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