Advertisement

CR Analytic Varieties with Given Boundary

  • Pierre Dolbeault
Part of the The University Series in Mathematics book series (USMA)

Abstract

Let us recall the Hartogs-Bochner-Martinelli theorem: let Ω be a bounded open set in ℂ n (n ≥ 2), with smooth connected boundary ∂Ω, and let f be a smooth CR function on ∂Ω. Then there exists FO(Ω) smooth on Ω̄ such that F | ∂Ω = f. The graph of f , gr f is a maximally complex submanifold of ℂ n+1 of dimension 2n — 1 ; gr(F | Ω) is a complex submanifold in ℂ n+1 \gr f of complex dimension n, and the boundary of gr̄ is gr f This gives a solution of a boundary problem in ℂ n+1 . We consider such boundary problems, with the solution given by geometric measure theory, so singularities of small measure will be allowed. Moreover, it will be convenient to consider linear combinations of subvarieties.

Keywords

Compact Support Boundary Problem Simple Extension Geometric Measure Theory Complex Submanifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. M. Chirka, Complex Analytic Sets, Kluwer Academic, Boston (1989); 1st edition in Russian, Nauka, Moscow (1985).zbMATHGoogle Scholar
  2. 2.
    P. Dolbeault, Sur les Chaînes Maximalement Complexes de Bord Donné, Proc. Symp. Pure Math., Vol. 44, pp. 171–205, American Mathematical Society, Providence, RI (1986).Google Scholar
  3. 3.
    R. Harvey, Holomorphic Chains and Their Boundaries, Proc. Symp. Pure Math., Vol 30, part 1, pp. 309–382, American Mathematical Society, Providence, RI (1977).Google Scholar
  4. 4.
    R. Harvey and J. Polking, Fundamental solution in complex analysis. I, Duke Math. J. 46, 253–300 (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    C. Laurent-Thiébaut, Résolution du ∂ à Support Compact et Phénomène de Hartogs-Bochner dans les Variétés CR, Proc. Symp. Pure Math., Vol. 52, part 3, pp. 239–249, American Mathematical Society, Providence, RI (1991).Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Pierre Dolbeault
    • 1
  1. 1.MathématiquesUniversité de Paris VI URA 213 du CNRSParis Cedex 05France

Personalised recommendations