Continuous Extension of Holomorphic Map**s

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Geometry of Holomorphic Map**s

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Abstract

In this chapter introduce the Kobayashi-Royden and other invariant metrics, and prove continuous extension to the boundary of proper holomorphic maps between strictly pseudoconvex domains.

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References

  1. Bedford, E., Pinchuk, S.: Convex domains with noncompact groups of automorphisms. Mat. Sb. 185, 3–26 (1994)

    MATH  Google Scholar 

  2. Berteloot, F.: Principe de Bloch et estimations de la metrique de Kobayashi des domaines de \(\mathbb C^2\). J. Geom. Anal. 13, 29–37 (2003)

    Google Scholar 

  3. Cho, S.: A lower bound on the Kobayashi metric near a point of finite type in \(\mathbb C^n\). J. Geom. Anal. 2, 317–322 (1992)

    Google Scholar 

  4. Diederich, K., Fornæss, J.E.: Proper holomorphic maps onto pseudoconvex domains with real analytic boundary. Ann. Math. 110, 575–592 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fornæss, J.E., Low, E.: Proper holomorphic map**s. Math. Scand. 58, 311–322 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fornæss, J.E., Sibony, N.: Construction of P.S.H. functions on weakly pseudoconvex domains. Duke Math. J. 58, 633–655 (1989)

    Google Scholar 

  7. Forstnerič, F., Rosay, J.P.: Localization of the Kobayashi metric and the boundary continuity of proper holomorphic map**s. Math. Ann. 279, 239–252 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Frankel, S.: Applications of affine geometry to geometric function theory in several complex variables. Part I. Convergent rescalings and intrinsic quasi-isometric structure. In: Several Complex Variables and Complex Geometry. Proceedings of Symposia in Pure Mathematics, vol. 52, part 2, pp. 183–208. American Mathematical Society, Providence (1991)

    Google Scholar 

  9. Graham, I.: Boundary behaviour of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in \(\mathbb C^n\) with smooth boundary. Trans. Am. Math. Soc. 207, 219–240 (1975)

    Google Scholar 

  10. Graham, I.: Sharp constants for the Koebe theorem and for estimates on intrinsic metrics on convex domains. In: Several Complex Variables and Complex Geometry. Proceedings of Symposia in Pure Mathematics, vol. 52, part 2, pp. 233–238. American Mathematical Society, Providence (1991)

    Google Scholar 

  11. Henkin, G.M.: An analytic polyhedron holomorphically nonequivalent to a strictly pseudoconvex domain. Sov. Math. Dokl. 14, 858–862 (1973)

    MATH  Google Scholar 

  12. Kobayashi, S.: Invariant distances on complex manifolds and holomorphic map**s. J. Math. Soc. Japan 19, 460–480 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kohn, J.: Supellipticity of the \(\overline \partial \)-Neumann problem on pseudoconvex domans: sufficient conditions. Acta Math. 142, 79–122 (1979)

    Google Scholar 

  14. Margulis, G.: Boundary correspondence under biholomorphic map**s between multi-dimensional domains. In: All Union Conference on the Theory of Functions, Harkov, pp. 137–138 (1971)

    Google Scholar 

  15. Pinchuk, S.: Proper holomorphic maps of strictly pseudoconvex domains. Sibirsk. Math. J. 15, 909–917 (1974)

    MathSciNet  Google Scholar 

  16. Royden, H.: Remarks on the Kobayashi Metric. Several Complex Variables II, Maryland, pp. 125–137. Springer, Berlin (1970/1971)

    Google Scholar 

  17. Sibony, N.: A Class of Hyperbolic Manifolds. Annals of Mathematics Studies, vol. 100, pp. 357–372. Princeton University Press, Princeton (1981)

    Google Scholar 

  18. Sukhov, A.: On the boundary regularity of holomorphic map**s. Mat. Sb. 185, 131–142 (1994)

    MATH  Google Scholar 

  19. Vormoor, N.: Topologische Fortsetzung biholomorpher Funktionen auf dem Rande bei beschränkten strengpseudokonvexen Gebieten im \(\mathbb C^n\)mit\(C^\infty \)-Rand. Math. Ann. 204, 239–261 (1973)

    Google Scholar 

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Pinchuk, S., Shafikov, R., Sukhov, A. (2023). Continuous Extension of Holomorphic Map**s. In: Geometry of Holomorphic Map**s. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-37149-3_3

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