Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with

Authors:
Emil J. Straube and Marcel K. Sucheston

Journal:
Trans. Amer. Math. Soc. **355** (2003), 143-154

MSC (2000):
Primary 32W05, 32T99; Secondary 53C12

Published electronically:
September 6, 2002

MathSciNet review:
1928081

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Abstract: Boas and Straube proved a general sufficient condition for global regularity of the -Neumann problem in terms of families of vector fields that commute approximately with . 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 E. Barrett,*Complex analytic realization of Reeb’s foliation of 𝑆³*, Math. Z.**203**(1990), no. 3, 355–361. MR**1038705**, 10.1007/BF02570743**2.**David E. Barrett,*Behavior of the Bergman projection on the Diederich-Fornæss worm*, Acta Math.**168**(1992), no. 1-2, 1–10. MR**1149863**, 10.1007/BF02392975**3.**Eric Bedford and John Erik Fornaess,*Domains with pseudoconvex neighborhood systems*, Invent. Math.**47**(1978), no. 1, 1–27. MR**0499316****4.**Eric Bedford and Bernard Gaveau,*Envelopes of holomorphy of certain 2-spheres in 𝐶²*, Amer. J. Math.**105**(1983), no. 4, 975–1009. MR**708370**, 10.2307/2374301**5.**Errett Bishop,*Differentiable manifolds in complex Euclidean space*, Duke Math. J.**32**(1965), 1–21. MR**0200476****6.**Harold P. Boas and Emil J. Straube,*Sobolev estimates for the \overline∂-Neumann operator on domains in 𝐶ⁿ admitting a defining function that is plurisubharmonic on the boundary*, Math. Z.**206**(1991), no. 1, 81–88. MR**1086815**, 10.1007/BF02571327**7.**Harold P. Boas and Emil J. Straube,*de Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the \overline∂-Neumann problem*, J. Geom. Anal.**3**(1993), no. 3, 225–235. MR**1225296**, 10.1007/BF02921391**8.**Harold P. Boas and Emil J. Straube: Global regularity of the -Neumann problem: a survey of the -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 Boston, Inc., Boston, MA, 1985. Translated from the Portuguese by Sue E. Goodman. MR**824240****10.**Alberto Candel and Lawrence Conlon,*Foliations. I*, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR**1732868****11.**John P. D’Angelo,*Several complex variables and the geometry of real hypersurfaces*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. MR**1224231****12.**G. B. Folland and J. J. Kohn,*The Neumann problem for the Cauchy-Riemann complex*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 75. MR**0461588****13.**Franc Forstnerič,*Complex tangents of real surfaces in complex surfaces*, Duke Math. J.**67**(1992), no. 2, 353–376. MR**1177310**, 10.1215/S0012-7094-92-06713-5**14.**André Haefliger,*Variétés feuilletées*, Ann. Scuola Norm. Sup. Pisa (3)**16**(1962), 367–397 (French). MR**0189060****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), no. 1, 1–21. MR**664323**, 10.1007/BF01393370**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**0433464****18.**Dennis Sullivan,*A foliation of geodesics is characterized by having no “tangent homologies”*, J. Pure Appl. Algebra**13**(1978), no. 1, 101–104. MR**508734**, 10.1016/0022-4049(78)90046-4**19.**Maria Athanassenas,*Volume-preserving mean curvature flow of rotationally symmetric surfaces*, Comment. Math. Helv.**72**(1997), no. 1, 52–66. MR**1456315**, 10.1007/PL00000366

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Additional Information

**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:
http://dx.doi.org/10.1090/S0002-9947-02-03133-1

Received by editor(s):
September 15, 2000

Published electronically:
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

Article copyright:
© Copyright 2002
American Mathematical Society