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Analytic discs in symplectic spaces

Published online by Cambridge University Press:  22 January 2016

Luca Baracco and
Giuseppe Zampieri
Luca Baracco
Affiliation:
Dipartimento di Matematica, Università di Padova via Belzoni 7 35131 Padova, Italy
Giuseppe Zampieri
Affiliation:
Dipartimento di Matematica, Università di Padova via Belzoni 7 35131 Padova, Italy
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Abstract

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We develop some symplectic techniques to control the behavior under symplectic transformation of analytic discs A of X = ℂn tangent to a real generic submanifold R and contained in a wedge with edge R.

We show that if A* is a lift of A to T* X and if χ is a symplectic transformation between neighborhoods of po and qo , then A is orthogonal to po if and only if Ã:= πχA* is orthogonal to qo . Also we give the (real) canonical form of the couples of hypersurfaces of ℝ2n ⋍ ℂn whose conormal bundles have clean intersection. This generalizes [10] to general dimension of intersection.

Combining this result with the quantized action on sheaves of the “tuboidal” symplectic transformation, we show the following: If R, S are submanifolds of X with R ⊂ S and then the conditions can be characterized as opposite inclusions for the couple of closed half-spaces with conormal bundles

In §3 we give some partial applications of the above result to the analytic hypoellipticity of CR hyperfunctions on higher codimensional manifolds by the aid of discs (cf. [2], [3] as for the case of hypersurfaces).

Keywords

32A4535C1532F10

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 161 , March 2001, pp. 55 - 67
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

References

[1] Baouendi, M. S., Rothshild, L. P. and Trepreau, J. M., On the geometry of analytic discs attached to real manifolds, J. Diff. Geometry, 39 (1994), 379–405.Google Scholar
[2] Baracco, L. and Zampieri, G., Analytic discs attached to half spaces of n and exten sion of holomorphic functions, J. Fac. Sci. Univ. Tokyo, (2001), to appear.Google Scholar
[3] Baracco, L. and Zampieri, G., Analytic discs attached to manifolds with boundary, Publ. R.I.M.S. Kyoto Univ., 33 (4) (1997), 687–694.Google Scholar
[4] Boggess, A., CR Manifolds and the tangential Cauchy-Riemann complex, Studies in Adv. Math. CRC Press, 1991.Google Scholar
[5] D’Agnolo, A. and Zampieri, G., Microlocal direct images of simple sheaves, Bull. Soc. Math. de France, 23 (1995), 101–133.Google Scholar
[6] Kashiwara, M. and Schapira, P., Microlocal study of sheaves, Astérisque Soc. Math. de France, 128 (1985).Google Scholar
[7] Trepreau, J. M., Sur le prolongement holomorphe des fonctions C-R définies sur une hypersurface réelle de classe C2 dans n , Invent. Math., 83 (1986), 583–592.Google Scholar
[8] Tumanov, A., Connections and propagation of analyticity for CR functions, Duke Math. Jour., 731 (1994), 1–24.Google Scholar
[9] Tumanov, A., Extending CR function from manifolds with boundaries, Math. Res. Letters, 2 (1995), 629–642.Google Scholar
[10] Zampieri, G., Simple sheaves along dihedral Lagrangians, Journal d’An. Math. de Jérusalem, 66 (1995), 331–344.Google Scholar
[11] Zampieri, G., Analytic discs attached to the conormal bundles of CR-manifolds, Advances in Math., 128 (1) (1997), 180–185.Google Scholar