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

Straube, Emil; Sucheston, Marcel

doi:10.1090/S0002-9947-02-03133-1

Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with $\bar \partial$
HTML articles powered by AMS MathViewer

by Emil J. Straube and Marcel K. Sucheston PDF

Trans. Amer. Math. Soc. 355 (2003), 143-154 Request permission

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.

References

David E. Barrett, Complex analytic realization of Reeb’s foliation of $S^3$, Math. Z. 203 (1990), no. 3, 355–361. MR 1038705, DOI 10.1007/BF02570743
David E. Barrett, Behavior of the Bergman projection on the Diederich-Fornæss worm, Acta Math. 168 (1992), no. 1-2, 1–10. MR 1149863, DOI 10.1007/BF02392975
Eric Bedford and John Erik Fornaess, Domains with pseudoconvex neighborhood systems, Invent. Math. 47 (1978), no. 1, 1–27. MR 499316, DOI 10.1007/BF01609476
Eric Bedford and Bernard Gaveau, Envelopes of holomorphy of certain $2$-spheres in $\textbf {C}^{2}$, Amer. J. Math. 105 (1983), no. 4, 975–1009. MR 708370, DOI 10.2307/2374301
Errett Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21. MR 200476
Harold P. Boas and Emil J. Straube, Sobolev estimates for the $\overline \partial$-Neumann operator on domains in $\textbf {C}^n$ admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), no. 1, 81–88. MR 1086815, DOI 10.1007/BF02571327
Harold P. Boas and Emil J. Straube, de Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the $\overline \partial$-Neumann problem, J. Geom. Anal. 3 (1993), no. 3, 225–235. MR 1225296, DOI 10.1007/BF02921391
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.
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, DOI 10.1007/978-1-4612-5292-4
Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
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
G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
Franc Forstnerič, Complex tangents of real surfaces in complex surfaces, Duke Math. J. 67 (1992), no. 2, 353–376. MR 1177310, DOI 10.1215/S0012-7094-92-06713-5
André Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 16 (1962), 367–397 (French). MR 189060
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, DOI 10.1007/BF01393370
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.
Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. MR 433464, DOI 10.1007/BF01390011
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, DOI 10.1016/0022-4049(78)90046-4
Maria Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric surfaces, Comment. Math. Helv. 72 (1997), no. 1, 52–66. MR 1456315, DOI 10.1007/PL00000366