Skip to main content
SpringerLink
Log in

Holomorphic Mappings and the Geometry of Hypersurfaces

  • Chapter
Introduction to Complex Analysis

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 7))

Abstract

The principal topic of this paper is non-degenerate (in the sense of Levi) hypersurfaces of complex manifolds and the automorphisms of such hypersur-faces. The material on strictly pseudoconvex hypersurfaces is presented most completely. We discuss in detail a form of writing the equations of the hyper-surface which allows one to carry out a classification of hypersurfaces. Certain biholomorphic invariants of hypersurfaces are considered. Especially, we consider in detail a biholomorphically invariant family of curves called chains. A lot of attention is given to constructing a continuation of a holomorphic mapping.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alexander, H.: Holomorphic mappings from the ball and polydisc. Math. Ann. 209, 249–256 (1974). Zbl. 272.32006 (Zbl. 281.32019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beloshapka, V.K.: On the dimension of the group of automorphisms of an analytic hypersurface. Izv. Akad. Nauk SSSR, Ser. Mat. 43, 243–266 (1979);

    MathSciNet  MATH  Google Scholar 

  3. Beloshapka, V.K.: On the dimension of the group of automorphisms of an analytic hypersurface. Math. USSR, Izv. 14, 223–245 (1980). Zbl. 412.58010

    Article  MATH  Google Scholar 

  4. Beloshapka, V.K.: An example of non-continuable holomorphic transformation of an analytic hypersurface. Mat. Zametki 32, 121–123 (1982);

    MathSciNet  Google Scholar 

  5. Beloshapka, V.K.: An example of non-continuable holomorphic transformation of an analytic hypersurface. Math. Notes 32, 540–541 (1983). Zbl. 518.32011

    Article  Google Scholar 

  6. Beloshapka, V.K., Vitushkin, A.G.: Estimates of the radius of convergence of power series defining a mapping of analytic hypersurfaces. Izv. Akad. Nauk SSSR, Ser. Mat. 45, 962–984 (1981);

    MathSciNet  MATH  Google Scholar 

  7. Beloshapka, V.K., Vitushkin, A.G.: Estimates of the radius of convergence of power series defining a mapping of analytic hypersurfaces. Math. USSR, Izv. 19, 241–259 (1982). Zbl. 492.32021

    Article  MATH  Google Scholar 

  8. Bochner, S.: Compact groups of differentiable transformations. Ann. Math., II. Ser. 46, 372–381 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  9. Burns, D., Diederich, K., Shnider, S.: Distinguished curves in pseudoconvex boundaries. Duke Math. J. 44, 407–31 (1977). Zbl. 382.32011

    Article  MathSciNet  MATH  Google Scholar 

  10. Burns, D., Shnider, S.: Real hypersurfaces in complex manifolds, Several complex variables. Proc. Symp. Pure Math. 30, Pt. 2, 141–168 (1977). Zbl. 422.32016

    MathSciNet  Google Scholar 

  11. Burns, D., Shnider, S.: Geometry of hypersurfaces and mapping theorems in Cn. Comment. Math. Helv. 54, No. 2, 199–217 (1979). Zbl. 444.32012

    Google Scholar 

  12. Burns, D., Shnider, S., Wells Jr., R.O.: Deformations of strictly pseudoconvex domains. Invent. Math. 46, 237–253 (1978). Zbl. 412.32022

    Article  MathSciNet  MATH  Google Scholar 

  13. Cartan, E.: Sur la géométrie pseudoconforme des hypersurfaces de deux variables complexes. Oeuvres complètes, Pt. II, Vol. 2, 1231–1304, Pt. III, Vol. 2, 1217–1238, Gauthier-Villars, Paris 1953 (Pt. II). Zbl. 58,83; 1955 (Pt. III). Zbl. 59,153

    Google Scholar 

  14. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974). Zbl. 302.32015

    Article  MathSciNet  Google Scholar 

  15. Ezhov, V.V.: Asymptotic behaviour of a strictly pseudoconvex surface along its chains. Izv. Akad. Nauk SSSR, Ser. Mat. 47, 856–880 (1983);

    MathSciNet  Google Scholar 

  16. Ezhov, V.V.: Asymptotic behaviour of a strictly pseudoconvex surface along its chains. Math. USSR, Izv. 23, 149–170 (1984). Zbl. 579.32034

    Article  Google Scholar 

  17. Ezhov, V.V., Kruzhilin, N.G., Vitushkin, A.G.: Extension of local mappings of pseudoconvex surfaces. Dokl. Akad. Nauk SSSR 270, 271–274 (1983);

    MathSciNet  Google Scholar 

  18. Ezhov, V.V., Kruzhilin, N.G., Vitushkin, A.G.: Extension of local mappings of pseudoconvex surfaces. Sov. Math., Dokl. 27, 580–583 (1983). Zbl. 558.32003

    MATH  Google Scholar 

  19. Ezhov, V.V., Kruzhilin, N.G., Vitushkin, A.G.: Extension of holomorphic maps along real-analytic hypersurfaces. Tr. Mat. Inst. Steklova 167, 60–95 (1985);

    MathSciNet  MATH  Google Scholar 

  20. Ezhov, V.V., Kruzhilin, N.G., Vitushkin, A.G.: Extension of holomorphic maps along real-analytic hypersurfaces. Proc. Steklov Inst. Math. 167, 63–102 (1986). Zbl. 575.32011

    MATH  Google Scholar 

  21. Faran, J.J.: Lewy’s curves and chains on real hypersurfaces. Trans. Am. Math. Soc. 265, 97–109 (1981). Zbl. 477.32021

    Article  MathSciNet  MATH  Google Scholar 

  22. Fefferman, C.L.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974). Zbl. 289.32012

    Article  MathSciNet  MATH  Google Scholar 

  23. Fefferman, C.L.: Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math., II. Ser. 103, 395–416 (1976). Zbl. 322.32012

    Article  MathSciNet  MATH  Google Scholar 

  24. Fefferman, C.L.: Parabolic invariant theory in complex analysis. Adv. Math. 31, 131–262 (1971). Zbl. 444.32013

    Article  MathSciNet  Google Scholar 

  25. Griffiths, P.A.: Two theorems on extensions of holomorphic mappings. Invent. Math. 14, 27–62 (1971). Zbl. 223.32016

    Article  MathSciNet  MATH  Google Scholar 

  26. Ivashkovich, S.M.: Extension of locally biholomorphic mappings of domains to a complex projective space. Izv. Akad. Nauk SSSR, Ser. Mat. 47, 197–206 (1983);

    MathSciNet  MATH  Google Scholar 

  27. Ivashkovich, S.M.: Extension of locally biholomorphic mappings of domains to a complex projective space. Math. USSR, Izv. 22, 181–189 (1984). Zbl. 523.32009

    Article  MATH  Google Scholar 

  28. Ivashkovich, S.M., Vitushkin, A.G.: Extension of holomorphic mappings of a real-analytic hypersurface to a complex projective space. Dokl. Akad. Nauk SSSR 267, 779–780 (1982);

    MathSciNet  Google Scholar 

  29. Ivashkovich, S.M., Vitushkin, A.G.: Extension of holomorphic mappings of a real-analytic hypersurface to a complex projective space. Sov. Math., Dokl. 26, 682–683 (1982). Zbl. 578.32024

    MATH  Google Scholar 

  30. Khenkin, G.M., Tumanov, A.E.: Local characterization of holomorphic automorphisms of Siegel domains. Funkts. Anal. Prilozh. 17, No. 4, 49–61 (1983);

    MathSciNet  Google Scholar 

  31. Khenkin, G.M., Tumanov, A.E.: Local characterization of holomorphic automorphisms of Siegel domains. Funct. Anal. Appl. 17, 285–294 (1983). Zbl. 572.32018

    MathSciNet  MATH  Google Scholar 

  32. Kruzhilin, N.G.: Estimation of the variation of the normal parameter of a chain on a strictly pseudoconvex surface. Izv. Akad. Nauk SSSR, Ser. Mat. 147, 1091–1113 (1983);

    MathSciNet  Google Scholar 

  33. Kruzhilin, N.G.: Estimation of the variation of the normal parameter of a chain on a strictly pseudoconvex surface. Math. USSR, Izv. 23, 367–389 (1984). Zbl. 579.32033

    Article  Google Scholar 

  34. Kruzhilin, N.G.: Local automorphisms and mappings of smooth strictly pseudoconvex hyper-surfaces. Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3, 566–591 (1985);

    MathSciNet  Google Scholar 

  35. Kruzhilin, N.G.: Local automorphisms and mappings of smooth strictly pseudoconvex hyper-surfaces. Math. USSR, Izv. 26, 531–552 (1986). Zbl. 597.32018

    Article  Google Scholar 

  36. Kruzhilin, N.G., Loboda, A.V.: Linearization of local automorphisms of pseudoconvex surfaces. Dokl. Akad. Nauk SSSR 271, 280–282 (1983);

    MathSciNet  Google Scholar 

  37. Kruzhilin, N.G., Loboda, A.V.: Linearization of local automorphisms of pseudoconvex surfaces. Sov. Math., Dokl. 28, 70–72 (1983). Zbl. 582.32040

    MATH  Google Scholar 

  38. Loboda, A.V.: On local automorphisms of real-analytic hypersurfaces. Izv. Akad. Nauk SSSR, Ser. Mat. 45, 620–645 (1981);

    MathSciNet  MATH  Google Scholar 

  39. Loboda, A.V.: On local automorphisms of real-analytic hypersurfaces. Math. USSR, Izv. 18, 537–559 (1982). Zbl. 473.32016

    Article  MATH  Google Scholar 

  40. Lewy, H.: On the boundary behaviour of holomorphic mappings. Acad. Naz. Lincei 35, 1–8 (1977)

    Google Scholar 

  41. Moser, J.K., Webster, S.M.: Normal forms for real surfaces in C2 near complex tangents, and hyperbolic surface transformations. Acta Math. 150, 255–296 (1983). Zbl. 519.32015

    Article  MathSciNet  MATH  Google Scholar 

  42. Pinchuk, S.I.: Holomorphic mappings of real-analytic hypersurfaces. Mat. Sb., Nov. Ser. 105, 574–593 (1978);

    MathSciNet  Google Scholar 

  43. Pinchuk, S.I.: Holomorphic mappings of real-analytic hypersurfaces. Math. USSR, Sb. 34, 503–519 (1978). Zbl. 389.32008

    Article  MATH  Google Scholar 

  44. Pinchuk, S.I.: A boundary uniqueness theorem for holomorphic functions of several complex variables. Mat. Zametki 75, 205–212 (1974);

    Google Scholar 

  45. Pinchuk, S.I.: A boundary uniqueness theorem for holomorphic functions of several complex variables. Math. Notes 75, 116–120 (1974). Zbl. 285.32002

    Article  MathSciNet  Google Scholar 

  46. Poincaré, H.: Les fonctions analytiques de deux variables et la représentation conforme. Rend. Circ. Mat. Palermo 23, 185–220 (1907)

    Article  MATH  Google Scholar 

  47. Rosay, J.-P.: Sur une caractérization de la boule parmi les domaines de Cn par son groupe d’automorphismes. Ann. Inst. Fourier 29, 91–97 (1979). Zbl. 402.32001 (Zbl. 414.32001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Segre, B.: Questioni geometriche legate colla teoría delle funzioni di due variabili complesse. Rend. Sem. Math. Roma, II. Ser. 7, No. 2, 59–107 (1982). Zbl. 5, 109

    Google Scholar 

  49. Shiftman, B.: Extension of holomorphic maps into hermitian manifolds. Math. Ann. 194, 249–258 (1971). Zbl. 213,360

    Article  MathSciNet  Google Scholar 

  50. Tanaka, N.J.: On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables. J. Math. Soc. Japan 14, 397–429 (1962). Zbl. 113,63

    MATH  Google Scholar 

  51. Tanaka, N.J.: On generalized graded Lie algebras and geometric structures. I. J. Math. Soc. Japan 19, 215–254 (1967). Zbl. 165,560

    MATH  Google Scholar 

  52. Vitushkin, A.G.: Holomorphic extension of mappings of compact hypersurfaces. Izv. Akad. Nauk SSSR, Ser. Mat. 46, 28–35 (1982);

    MathSciNet  Google Scholar 

  53. Vitushkin, A.G.: Holomorphic extension of mappings of compact hypersurfaces. Math. USSR, Izv. 20, 27–33 (1983). Zbl. 571.32011

    Article  MATH  Google Scholar 

  54. Vitushkin, A.G.: Global normalization of a real-analytic surface along a chain. Dokl. Akad. Nauk SSSR 269, 15–18 (1983);

    MathSciNet  Google Scholar 

  55. Vitushkin, A.G.: Global normalization of a real-analytic surface along a chain. Sov. Math., Dokl. 27, 270–273 (1983). Zbl. 543.32003

    MATH  Google Scholar 

  56. Vitushkin, A.G.: Analysis of power series defining automorphisms of hypersurfaces in connection with the problem of extending maps. International Conf. on analytic methods in the theory of numbers and analysis Moscow, 14–19 September 1981, Tr. Mat. Inst. Steklova 163, 37–41 (1984);

    Google Scholar 

  57. Vitushkin, A.G.: Analysis of power series defining automorphisms of hypersurfaces in connection with the problem of extending maps. Proc. Steklov Inst. Math. 163, 47–51 (1985). Zbl. 575.32010

    Google Scholar 

  58. Vitushkin, A.G.: Real-analytic hypersurfaces in complex manifolds. Usp. Mat. Nauk 40, No. 2, 3–31 (1985);

    MathSciNet  Google Scholar 

  59. Vitushkin, A.G.: Real-analytic hypersurfaces in complex manifolds. Russ. Math. Surv. 40, No. 2, 1–35 (1985). Zbl. 588.32025

    Article  MathSciNet  MATH  Google Scholar 

  60. Webster, S.M.: Kähler metrics associated to a real hypersurface. Comment. Math. Helv. 52, 235–250 (1977). Zbl. 354.53050

    Article  MathSciNet  MATH  Google Scholar 

  61. Webster, S.M.: Pseudo-hermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1978). Zbl. 379.53016 (Zbl. 394.53023)

    MathSciNet  MATH  Google Scholar 

  62. Webster, S.M.: On the Moser normal form at a non-umbilic point. Math. Ann. 233, 97–102 (1978). Zbl. 358.32013

    Article  MathSciNet  MATH  Google Scholar 

  63. Wells Jr., R.O.: On the local holomorphic hull of a real submanifold in several complex variables. Commun. Pure Appl. Math. 19, 145–165 (1966). Zbl. 142,339

    Article  MATH  Google Scholar 

  64. Wells Jr., R.O.: Function theory on differentiable submanifolds. Contributions to analysis. Academic Press, 407–441, New York 1974. Zbl. 293.32001

    Google Scholar 

  65. Wells Jr., R.O.: The Cauchy-Riemann equations and differential geometry. Bull. Amer. Math. Soc. 6, 187–199 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  66. Wong, B.: Characterization of the unit ball in C by its automorphism group. Invent. Math. 41, 253–257 (1977).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. A. G. Vitushkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Vitushkin, A.G. (1997). Holomorphic Mappings and the Geometry of Hypersurfaces. In: Introduction to Complex Analysis. Encyclopaedia of Mathematical Sciences, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61525-2_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-61525-2_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63005-0

  • Online ISBN: 978-3-642-61525-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation