Transactions of the American Mathematical Society

Author(s): Franc Forstneric; Christine Laurent-Thiébaut
Journal: Trans. Amer. Math. Soc. 360 (2008), 307-329.
MSC (2000): Primary 32D15, 32T20, 32T27, 32V05, 32V25; Secondary 57R30
Posted: July 23, 2007
Retrieve article in: PDF DVI PostScript

Abstract: We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface

and holomorphic convexity of compact sets in

, or bounded in part by

. Applications include extendability of Cauchy-Riemann functions, solvability of the $\overline{\partial}_b$ -equation, approximation of Cauchy-Riemann and holomorphic functions, and global regularity of the $\overline{\partial}$ -Neumann operator.

1.: AHERN, P., RUDIN, W., Hulls of -spheres in $\mathbb{C}^3$ , in The Madison Symposium on Complex Analysis (Madison, WI, 1991, 1-27), Contemp. Math., 137, Amer. Math. Soc., Providence, RI, 1992. MR 1190966 (93k:32020)
2.: AIRAPETIAN, R. A., Extending CR functions from piecewise smooth CR manifolds. Math. Sb., 134 (1987), 108-118 (Russian).
3.: ANDREOTTI, A., NACINOVICH, M., Analytic Convexity. Ann. Sc. Norm. Sup. Pisa, 7 (1980), 287-372. MR 581145 (81m:32025)
4.: AUDIN, M., Fibrés normaux d'immersions en dimension double, points doubles d'immersions lagrangiennes et plongements totalement réels. Comment. Math. Helv., 63 (1988), 593-623. MR 966952 (89m:57032)
5.: BAOUENDI, M. S., TRÈVES, F., A property of the functions and distributions annihilated by a locally integrable system of complex vector fields. Ann. of Math., (2) 113 (1981), 387-421. MR 607899 (82f:35057)
6.: BARRETT, D. E., Complex analytic realization of Reeb's foliation of $S\sp 3$ . Math. Z., 203 (1990), 355-361. MR 1038705 (91f:32018)
7.: -, Global convexity properties of some families of three-dimensional compact Levi-flat hypersurfaces. Trans. Amer. Math. Soc., 332 (1992), 459-474. MR 1055805 (93c:32026)
8.: -, Behavior of the Bergman projection on the Diederich-Fornæss worm. Acta Math., 168 (1992), 1-10. MR 1149863 (93c:32033)
9.: BARRETT, D. E., FORNÆSS, J.-E., On the smoothness of Levi-foliations. Publ. Mat., 32 (1988), 171-177. MR 975896 (90b:32037)
10.: BEDFORD, E., DE BARTOLOMEIS, P., Levi flat hypersurfaces which are not holomorphically flat. Proc. Amer. Math. Soc., 81 (1981), 575-578. MR 601733 (82a:32025)
11.: BEDFORD, E., FORNÆSS, J. E., Domains with pseudoconvex neighborhood systems. Invent. Math., 47 (1978), 1-27. MR 0499316 (58:17215)
12.: BOAS, H. P., STRAUBE, E., Sobolev estimates for the $\overline{\partial}$ -Neumann operator on domains in $\mathbb{C}^n$ admitting a defining function that is plurisubharmonic at the boundary. Math. Z., 206 (1991), 81-88. MR 1086815 (92b:32027)
13.: -, 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), 225-235. MR 1086815 (92b:32027)
14.: BOGGESS, A., CR Manifolds and the Tangential Cauchy-Riemann Complex. CRC Press, Boca Raton, 1991. MR 1211412 (94e:32035)
15.: CAMACHO, C., LINS NETO, A., Geometric Theory of Foliations. Birkhäuser, Boston, 1985. MR 824240 (87a:57029)
16.: CANDEL, A., CONLON, L., Foliations I. Grad. Studies in Math., 23, Amer. Math. Soc., Providence, Rhode Island, 2000. MR 1732868 (2002f:57058)
17.: CHEN, S.-C., SHAW, M.-C., Partial Differential Equations in Several Complex Variables. Amer. Math. Soc. and International Press, Providence, RI, 2001. MR 1800297 (2001m:32071)
18.: CHIRKA, E. M., Analytic representation of CR functions. (Russian) Math. USSR Sbornik, 27 (1975), 526-553.
19.: CHRIST, M., Global $\mathbb{C}^\infty$ irregularity of the $\overline{\partial}$ -Neumann problem for worm domains. J. Amer. Math. Soc., 9 (1996), 1171-1185. MR 1370592 (96m:32014)
20.: COLTOIU, M., Complete locally pluripolar sets. J. Reine Angew. Math., 412 (1990), 108-112. MR 1074376 (91h:32010)
21.: D'ANGELO, J. P., Real hypersurfaces, orders of contact, and applications. Annals of Math., 115 (1982), 615-637. MR 657241 (84a:32027)
22.: DEMAILLY, J.-P., Cohomology of -convex spaces in top degrees. Math. Z., 204 (1990), 283-295. MR 1055992 (91e:32014)
23.: DIEDERICH, K., FORNÆSS, J.-E., Pseudoconvex domains: An example with nontrivial Nebenhülle. Math. Ann., 225 (1977), 275-292. MR 0430315 (55:3320)
24.: FOLLAND, G. B., KOHN, J. J., The Neumann Problem for the Cauchy-Riemann Complex. Annals of Math. Studies 75, Princeton Univ. Press, 1972. MR 0461588 (57:1573)
25.: FORNÆSS, J. E., NAGEL, A., The Mergelyan property for weakly pseudoconvex domains. Manuscripta Math., 22 (1977), 199-208. MR 0457779 (56:15983)
26.: FORSTNERIC, F., On totally real embeddings into $\mathbb{C}^n$ . Expo. Math., 4 (1986), 243-255. MR 880125 (88g:32016)
27.: -, A contractible Levi-flat hypersurface in $\mathbb{C}^2$ which is a determining set for pluriharmonic functions. Arkiv Math. 44 (2006), 87-91. MR 2237212 (2007c:32047)
28.: GHYS, É., L'invariant de Godbillon-Vey. Astérisque, 177-178 (1989), 155-181. MR 1040572 (91h:57015)
29.: GIGANTE, G., TOMASSINI, G., Foliations with complex leaves. Diff. Geom. Appl., 5 (1995), 33-49. MR 1319934 (96b:32024)
30.: GODBILLON, C., Dynamical Systems on Surfaces. Springer-Verlag, Berlin, 1983. MR 681119 (84b:57018)
31.: -, Feuilletages, études géométriques. Birkhäuser, Basel-Boston-Berlin, 1991. MR 1120547 (93i:57038)
32.: GODBILLON, C., VEY, J., Un invariant des feuilletages de codimension 1. C. R. Acad. Sci. Paris, Sér. A, 273 (1971), 92-95. MR 0283816 (44:1046)
33.: GROMOV, M., Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenz- gebiete (3), 9. Springer, Berlin-New York, 1986. MR 864505 (90a:58201)
34.: GUNNING, R. C., ROSSI, H., Analytic Functions of Several Complex Variables. Prentice-Hall, Englewood Cliffs, NJ, 1965. MR 0180696 (31:4927)
35.: HAEFLIGER, A., Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes. Comment. Math. Helv., 32 (1958), 248-329. MR 0100269 (20:6702)
36.: HAEFLIGER, A., REEB, G., Variétés (non séparées) à un dimension et structures feuilletées du plan. L'Enseignement Math., 3 (1957), 107-125. MR 0089412 (19:671c)
37.: HENKIN, G. M., LEITERER, J., Theory of functions on complex manifolds. Akademie-Verlag, Berlin, 1984. MR 774049 (86a:32002)
38.: HÖRMANDER, L. An Introduction to Complex Analysis in Several Variables, 3rd ed. North Holland, Amsterdam, 1990. MR 1045639 (91a:32001)
39.: KAMKE, E., Über die partielle Differentialgleichung . Math. Z., 41 (1936), 56-66; 42 (1936), 287-300. MR 1545604
40.: KOHN, J. J., A survey of the $\overline{\partial}$ -Neumann problem. Proc. Symp. Pure Math., 41, pp. 137-145. Amer. Math. Soc., Providence, R.I., 1984. MR 740877 (85e:32023)
41.: LAURENT-THIÉBAUT, C., Sur l'extension des fonctions CR dans une variété de Stein. Ann. Mat. Pura Appl., 150 (1988), 141-151. MR 946033 (89j:32020)
42.: -, Extension de formes différentielles CR. C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 539-542. MR 0941620 (89e:32039)
43.: -, Sur l'équation de Cauchy-Riemann tangentielle dans une calotte strictement pseudoconvexe. Internat. J. Math., 16 (2005), 1063-1079. MR 2180065 (2006m:32044)
44.: LAURENT-THIÉBAUT, C., LEITERER, J., On the Hartogs-Bochner extension phenomenon for differential forms. Math. Ann., 284 (1989), 103-119. MR 995385 (90c:32026)
45.: LAURENT-THIÉBAUT, C., PORTEN, E., Analytic extension from non-pseudoconvex boundaries and -convexity. Ann. Inst. Fourier (Grenoble), 53 (2003), 847-857. MR 2008443 (2004h:32041)
46.: LUPACCIOLU, G., Characterization of removable sets in strongly pseudoconvex boundaries. Arkiv Math., 32 (1994), 455-473. MR 1318542 (96a:32027)
47.: LUPACCIOLU, G., TOMASSINI, G., An extension theorem for CR-functions. Ann. Mat. Pura Appl., 137 (1984), 257-263. MR 772261 (86e:32021)
48.: MASA, X., Alexander-Spanier cohomology of foliated manifolds. Illinois J. Math., 46 (2002), 979-998. MR 1988246 (2004c:57043)
49.: NEMIROVSKI, S., Stein domains with Levi-flat boundaries on compact complex surfaces. Mat. Zametki, 66 (1999), 632-634. English transl.: Math. Notes, 66 (1999), 522-525. MR 1747093 (2001d:32025)
50.: NOVIKOV, S. P., Topology of foliations. Trans. Moscow Math. Soc., 14 (1965), 268-305. MR 0200938 (34:824)
51.: OHSAWA, T., On the Levi-flats in complex tori of dimension two. Publ. Res. Inst. Math. Sci. 42 (2006), 361-377. MR 2250063
52.: ROSSI, H., Holomorphically convex sets in several complex variables. Annals Math., 24 (1961), 470-493. MR 0133479 (24:A3310)
53.: SACKSTEDER, R., Foliations and pseudogroups. Amer. J. Math., 87 (1965), 79-102. MR 0174061 (30:4268)
54.: SHAW, M.-C., $L\sp p$ estimates for local solutions of $\overline\partial\sb {\rm b}$ on strongly pseudo-convex CR manifolds. Math. Ann., 288 (1990), 35-62. MR 1070923 (92b:32028)
55.: -, Local existence theorems with estimates for $\overline\partial\sb b$ on weakly pseudo-convex CR manifolds. Math. Ann., 294 (1992), 677-700. MR 1190451 (94b:32026)
56.: SHCHERBINA, N. V., Decomposition of a common boundary of two domains of holomorphy into analytic curves. Iszvestia Akad. Nauk SSSR Ser. Mat, 46 (1982), 1106-1123. English transl.: Math. USSR Iszvestia, 21 (1983), 399-413. MR 675533 (84b:32018)
57.: SIU, Y.-T., Every Stein subvariety admits a Stein neighborhood. Invent. Math., 38 (1976), 89-100. MR 0435447 (55:8407)
58.: -, $\overline{\partial}$ -regularity for weakly pseudoconvex domains in compact Hermitian symmetric spaces with respect to invariant metrics. Ann. Math., (2) 156 (2002), 595-621. MR 1933078 (2003h:32057)
59.: SLODKOWSKI, Z., Analytic set-valued functions and spectra. Math. Ann., 256 (1981), 363-386. MR 626955 (83b:46070)
60.: STRAUBE, E. J., A sufficient condition for global regularity of the $\overline{\partial}$ -Neumann operator. Preprint, 2005. [arXiv:math.CV/0510354]
61.: -, Aspects of the -Sobolev theory of the $\overline{\partial}$ -Neumann problem. Proceedings of the International Congress of Mathematicians, Madrid, 2006.
62.: STRAUBE, E. J., SUCHESTON, M. K., Plurisubharmonic defining functions, good vector fields, and exactness of a certain one-form. Monatsh. Math., 136 (2002), 249-258. MR 1919648 (2003f:32051)
63.: -, Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with $\overline{\partial}$ . Trans. Amer. Math. Soc., 355 (2002), 143-154. MR 1928081 (2003h:32058)
64.: SULLIVAN, D., Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math., 36 (1976), 225-255. MR 0433464 (55:6440)
65.: TONDEUR, P., Geometry of Foliations. Birkhäuser, Boston, 1997. MR 1456994 (98d:53037)
66.: WAZEWSKY, T., Sur l'équation . Mathematica, 8 (1934), 103-116; 9 (1935), 179-182.

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32D15, 32T20, 32T27, 32V05, 32V25, 57R30

Franc Forstneric
Affiliation: Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Email: franc.forstneric@fmf.uni-lj.si

Christine Laurent-Thiébaut
Affiliation: Institut Fourier, UMR 5582 CNRS/UJF, BP 74, 38402 St. Martin d'Hères Cedex, France
Email: Christine.Laurent@ujf-grenoble.fr

DOI: 10.1090/S0002-9947-07-04263-8
PII: S 0002-9947(07)04263-8
Keywords: Levi-flat hypersurfaces, foliations, Stein manifolds
Received by editor(s): March 17, 2005
Received by editor(s) in revised form: February 7, 2006
Posted: July 23, 2007
Additional Notes: The first author was supported by grants P1-0291 and J1-6173, Republic of Slovenia.
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			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