Transactions of the American Mathematical Society

Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with $\bar\partial$

Author(s): Emil J. Straube; Marcel K. Sucheston
Journal: Trans. Amer. Math. Soc. 355 (2003), 143-154.
MSC (2000): Primary 32W05, 32T99; Secondary 53C12
Posted: September 6, 2002
Retrieve article in: PDF DVI PostScript

Abstract: Boas and Straube proved a general sufficient condition for global regularity of the $\bar\partial$ -Neumann problem in terms of families of vector fields that commute approximately with $\bar\partial$ . In this paper, we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds. This question turns out to be closely related to properties of interest from the point of view of foliation theory.

1.: David Barrett: Complex analytic realization of Reeb's foliation of , Math. Z. 203 (1990), 355-361. MR 91f:32018
2.: David Barrett: Behavior of the Bergman projection on the Diederich-Fornaess worm, Acta Math. 168, 1-2 (1992), 1-10. MR 93c:32033
3.: Eric Bedford and John Erik Fornaess: Domains with pseudoconvex neighborhood systems, Invent. Math. 47 (1978), 1-27. MR 58:17215
4.: Eric Bedford and Bernard Gaveau: Envelopes of holomorphy of certain 2-spheres in $\mathbb{C} ^2$ , American J. of Math. 105 (1983), 975-1009. MR 84k:32016
5.: Errett Bishop: Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1-21. MR 34:369
6.: Harold P. Boas and Emil J. Straube: Sobolev estimates for the $\bar\partial$ -Neumann operator on domains in $\mathbb{C} ^n$ admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), 81-88. MR 92b:32027
7.: Harold P. Boas and Emil J. Straube: De Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the $\bar\partial$ -Neumann problem, J. Geometric Analysis 3, Nr. 3 (1993), 225-235. MR 94g:32019
8.: Harold P. Boas and Emil J. Straube: Global regularity of the $\bar\partial$ -Neumann problem: a survey of the $\mathcal{L}^2$ -Sobolev theory, Several Complex Variables, M. Schneider and Y.-T. Siu editors, Mathematical Sciences Research Institute Publications 37, 79-111, Cambridge Univ. Press, 1999.
9.: César Camacho and Alcides Lins Neto: Geometric Theory of Foliations, Birkhäuser, 1985. MR 87a:57029
10.: Alberto Candel and Lawrence Conlon: Foliations I, Graduate Studies in Mathematics, vol. 23, American Math. Society, 2000. MR 2002f:57058
11.: John P. D'Angelo: Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, 1993. MR 94i:32022
12.: G. B. Folland and J. J. Kohn: The Neumann Problem for the Cauchy-Riemann Complex, Annals of Math. Studies 75, Princeton Univ. Press, 1972. MR 57:1573
13.: Franc Forstneric: Complex tangents of real surfaces in complex surfaces, Duke Math. J. 67 (1992), 353-376. MR 93g:32025
14.: André Häfliger: Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa 16 (1962), 367-397. MR 32:6487
15.: Carlos E. Kenig and Sidney M. Webster: The local hull of holomorphy of a surface in the space of two complex variables, Invent. Math. 67 (1982), 1-21. MR 84c:32014
16.: Emil J. Straube and Marcel K. Sucheston: Plurisubharmonic defining functions, good vector fields, and exactness of a certain one form, Monatsh. Math. 136 (2002), 249-258.
17.: Dennis Sullivan: Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225-255. MR 55:6440
18.: Dennis Sullivan: A foliation of geodesics is characterized by having no ``tangent homologies'', J. Pure and Applied Algebra 13 (1978), 101-104. MR 80c:57019
19.: Philippe Tondeur: Geometry of Foliations, Birkhäuser, 1997. MR 98d:58037

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32W05, 32T99, 53C12

Emil J. Straube
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: straube@math.tamu.edu

Marcel K. Sucheston
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

DOI: 10.1090/S0002-9947-02-03133-1
PII: S 0002-9947(02)03133-1
Received by editor(s): September 15, 2000
Posted: September 6, 2002
Additional Notes: Research supported in part by NSF grant DMS-9801539
Marcel K. Sucheston died tragically on April 24, 2000. The surviving author dedicates this paper to his memory
Copyright of article: Copyright 2002, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS