Mathematische Annalen

, Volume 332, Issue 1, pp 55–65 | Cite as

Smooth submanifolds intersecting any analytic curve in a discrete set

  • Dan Coman
  • Norman Levenberg
  • Evgeny A. PoletskyEmail author


We construct examples of C smooth submanifolds in ℂn and ℝn of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianalytic functions. In the complex case, these submanifolds contain real n-dimensional tori or Euclidean spaces that are not pluripolar while the intersection with any complex analytic disk is polar.


Euclidean Space Analytic Curve Complex Case Smooth Submanifolds Analytic Disk 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bedford, E.: Survey of pluripotential theory. In: J.-E. Fornæss (ed.) Several Complex Variables: Proceedings of the Mittag-Leffler Institut 1987-1988. Math. Notes 38, Princeton Univ. Press, 48–97 (1993)Google Scholar
  2. 2.
    Bierstone, E., Milman, P.: Arc-analytic functions. Invent. Math. 101, 411–424 (1990)Google Scholar
  3. 3.
    Bierstone, E., Milman, P.: Resolution of singularities in Denjoy–Carleman classes. Selecta Math. (N.S.) 10, 1–28 (2004)CrossRefGoogle Scholar
  4. 4.
    Coman, D., Levenberg, N., Poletsky, E.A.: Quasianalyticity and pluripolarity. (preprint)Google Scholar
  5. 5.
    Diederich, K., Fornæss, J.E.: Pseudoconvex domains with real-analytic boundary. Annals of Math. (2) 107, 371–384 (1978)Google Scholar
  6. 6.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer-Verlag, 1990Google Scholar
  7. 7.
    Kiselman, C.O.: The growth of restriction of plurisubharmonic functions, Math. Analysis and Applications. Part B. Adv. Math. Suppl. Studies 7B, 435–454 (1981)Google Scholar
  8. 8.
    Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Birkhäuser, 1992Google Scholar
  9. 9.
    Mandelbrojt, S.: Séries de Fourier et classes quasi-analytique de fonctions. Paris, Gauthier-Villars, 1935Google Scholar
  10. 10.
    Pinčuk, S.I.: A boundary uniqueness theorem for holomorphic functions of several complex variables. Mat. Zametki 15, 205–212 (1974)Google Scholar
  11. 11.
    Ronkin, L.I.: Introduction to the theory of entire functions of several variables. Amer. Math. Soc. Providence, R.I. 1974Google Scholar
  12. 12.
    Sadullaev, A.: A boundary uniqueness theorem in ℂn. Mat. Sb. (N.S.) 101(143), 568–583 (1976)Google Scholar
  13. 13.
    Timan, A.F.: Theory of Approximation of Functions of a Real Variable. Pergamon Press, Macmillan, New York, 1963Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Dan Coman
    • 1
  • Norman Levenberg
    • 2
  • Evgeny A. Poletsky
    • 1
    Email author
  1. 1.Department of MathematicsSyracuse UniversitySyracuseUSA
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA

Personalised recommendations