On the defect of an analytic disc
Although the concept of defect of an analytic disc attached to a generic manifold of Cn seems to play a merely technical role, it turns out to be a rather deep and fruitful notion for the extendability of CR functions defined on the manifold. In this paper we give a new geometric description of defect, drawing attention to the behaviour of the interior points of the disc by infinitesimal perturbations. For hypersurfaces a stronger result holds because these perturbations describe a complex vector space of Cn. For a big analytic disc the defect does not need to be smaller than the codimension of the manifold. Indeed we show by an example that it can be arbitrarily large independently of the codimension of the manifold. Nevertheless we also prove that the defect is always finite. In the case of a hypersurface we give a geometric upper bound for the defect.
KeywordsVector Space Interior Point Complex Vector Strong Result Geometric Description
Unable to display preview. Download preview PDF.
- [BR]M. S. Baouendi -L. P. Rotschild,A generalized complex Hopf lemma and its applications to CR mappings, Invent. Math.,111 (1993), pp. 331–348.Google Scholar
- [BRT]M. S.Baouendi - L. P.Rotschild - J. M.Trépreau,On the geometry of analytic discs attached to real manifolds, J. Diff. Geom. (to appear).Google Scholar
- [CH]R. Courant -D. Hilbert,Methods of Mathematical Physics, Vol. 2, Interscience Publisher, New York (1962).Google Scholar
- [CR]E. M. Chirka -C. Rea,Normal and tangent ranks of CR mappings, Duke Math. J.,76, N. 2 (1994), pp. 417–431.Google Scholar
- [R]P.Rossi,Estensione in un wedge di funzioni CR, Tesi di laurea, Univ. di Roma «Tor Vergata» (1991).Google Scholar
- [T1]A. E. Tumanov,Extension of CR functions into a wedge from a manifold of finite type, Math. Sbornik,178 (1988), pp. 128–139 (English Transl. in Math. USSR Sbornik,64 (1989), pp. 129–140).Google Scholar
- [T2]A. E. Tumanov,Extension of CR functions into a wedge, Math. Sbornik,181 (1990), pp. 385–398 (English Transl. in Math. USSR Sbornik,70 (1991), pp. 385–398).Google Scholar
- [Tr]J. M. Trépreau,Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. Fr.,118 (1990), pp. 403–450.Google Scholar
- [V]N. P. Vekua,Systems of Singular Integral Equations, P. Noordhoff Ltd., Groningen, The Netherlands (1967).Google Scholar