Abstract
The authors introduce a Dirichlet integral-type biholo-morphic-invariant pseudodistance connected with bordered holomorphic chains of dimension one whose regular part is treated as a Riemann surface. The condition for a complex manifold that the pseudodistance on it is a distance defines a class of hyperbolic-like manifolds which have an important property of extendability of holomorphic mappings, analogous to the hyperbolic manifolds, Stein spaces, and complex spaces with a Stein covering.
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References
ANDREOTTI, A. and J. ŁAWRYNOWICZ: ‘On the generalized complex Monge-Ampère equation on complex manifolds and related questions’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 943–948.
ANDREOTTI, A. and W. STOLL: ‘Extension of holomorphic maps’, Ann. of Math. (2) 72 (1960), 312–349.
BESNAULT, J. et P. DOLBEAULT: ‘Sur les bords d’ensembles analytiques complexes dans IPn(C)’, Sympos. Math. 24 (1981), 205–213.
CHERN, S.S., H.I. LEVINE, and L. NIRENBERG: ‘Intrinsic norms on a complex manifold’, in Global analysis, Papers in honor of K. Kodaira, ed. by D.C. Spencer and S. Iynaga, Univ. of Tokyo Press and Princeton Univ. Press, Tokyo 1969, pp. 119–139;
CHERN, S.S., H.I. LEVINE, and L. NIRENBERG: ‘Intrinsic norms on a complex manifold’ reprinted in S.S. Chern: Selected papers, Springer, New York-Heidelberg-Berlin 1978, pp. 371–391.
DOLBEAULT, P.: ‘On holomorphic chains with given boundary in IPn (⊄)’, in Analytic functions, Błażejewko 1982, Proceedings, ed. by J. Ławrynowicz (Lecture Notes in Math. 1039), Springer, Berlin--Heidelberg-New York-Tokyo 1983, pp. 118–129.
DOLBEAULT, P.: ‘Sur les chaines maximalement complexes de bord donné’, Proc. Sympos. Pure Math. 44 (1986), 171–205.
GAVEAU, B. et J. ŁAWRYNOWICZ: ‘Espaces de Dirichlet invariants biholomorphes et capacités associées’, Bull. Acad. Volon. Sci. Sér. Sci. Math. 30 (1982), 63–69.
GAVEAU, B. et J. ŁAWRYNOWICZ: ‘intégrale de Dirichlet sur une variété complexe I’, in Séminaire Pierre Le long — Henri Skoda (Analyse), Années 1980/81 (Lecture Notes in Math. 919), Springer, Berlin-Heidelberg-New York 1982, pp. 131–151
GAVEAU, B. et J. ŁAWRYNOWICZ: ‘intégrale de Dirichlet sur une variété complexe I’, in Séminaire Pierre Le long — Henri Skoda (Analyse), Années 1980/81 (Lecture Notes in Math. 919), Springer, Berlin-Heidelberg-New York 1982, 163–165.
HARVEY, R.: ‘Holomorphic chains and their boundaries’, Proc. Sympos. Pure Math. 30, 1 (1977), 309–382.
HARVEY, R. and B. LAWSON: ‘On boundaries of complex analytic varieties I-II’, Ann. of Math. (2) 102 (1975), 233–290.
HARVEY, R. and B. LAWSON: ‘On boundaries of complex analytic varieties I-II’, Ann. of Math.(2) 106 (1977), 213–238.
KOBAYASHI, S.: ‘invariant distances on complex manifolds and holomorphic mappings’, J. Math. Soc. Japan 19 (1967), 460–480.
KOBAYASHI, S.: Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, Inc., New York 1970.
KOBAYASHI, S.: ‘Intrinsic distances, measures and geometric function theory’, Bull. Amer. Math. Soc. 82 (1976), 357–416.
ŁAWRYNOWICZ, J.: ‘Condenser capacities and an extension of Schwarz’s lemma for hermitian manifolds’, Bull. Acad. Polon. Sci. Sér. Sci, Math. Astronom. Phys. 23 (1975), 839–844.
ŁAWRYNOWICZ, J.: ‘On a class of capacities on complex manifolds endowed with an hermitian structure and their relation to elliptic and hyperbolic quasiconformal mappings’, (a) Conf. Analytic Functions Abstracts, Cracow 1974, pp. 33–34 (abstract),
ŁAWRYNOWICZ, J.: Ann. Polon. Math. 33 (1976), 178 (abstract),
ŁAWRYNOWICZ, J.: Dissertationes Math. 166 (1980), 48 pp. (in extenso).
ŁAWRYNOWICZ, J. and M. OKADA: ‘Canonical diffusion and foliation involving the complex hessian’, (a) Inst, of Math. Polish Acad. Sci. Preprint no. 356 (1985), II + 10 pp.,
ŁAWRYNOWICZ, J. and M. OKADA, Bull. Polish Acad. Sci. Math. 34 (1986), 661–667.
SCHIFFMAN, B.: ‘On the removal of singularities of analytic sets’, Michigan Math. J. 15 (1968), 111–120.
SERRE, J.-P.: ‘Géométrie algébrique et géométrie analytique’, Ann. Inst. Fourier 6 (1956), 1–41.
STOLL, W.: ‘Über meromorphe Abbildungen komplexer Räume I-II’, Math. Ann. 136 (1958), 201–239
STOLL, W.: ‘Über meromorphe Abbildungen komplexer Räume I-II’, Math. Ann. 136 (1958), 393–429.
THULLEN, P.: ‘Über die wesentlichen Singularitäten analytischer Funktionen und Flächen im Räume von n komplexen Veränderlichen’, Math. Ann. 111 (1935), 137–157.
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Dolbeault, P., Ławrynowicz, J. (1989). Holomorphic Chains and Extendability of Holomorphic Mappings. In: Ławrynowicz, J. (eds) Deformations of Mathematical Structures. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2643-1_18
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DOI: https://doi.org/10.1007/978-94-009-2643-1_18
Publisher Name: Springer, Dordrecht
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