Stein compacts in Levi-flat hypersurfaces

Forstnerič, Franc; Laurent-Thiébaut, Christine

doi:10.1090/S0002-9947-07-04263-8

Stein compacts in Levi-flat hypersurfaces
HTML articles powered by AMS MathViewer

by Franc Forstnerič and Christine Laurent-Thiébaut PDF

Trans. Amer. Math. Soc. 360 (2008), 307-329 Request permission

Abstract:

We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface $M$ and holomorphic convexity of compact sets in $M$, or bounded in part by $M$. 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.

References

Patrick Ahern and Walter Rudin, Hulls of $3$-spheres in $\textbf {C}^3$, The Madison Symposium on Complex Analysis (Madison, WI, 1991) Contemp. Math., vol. 137, Amer. Math. Soc., Providence, RI, 1992, pp. 1–27. MR 1190966, DOI 10.1090/conm/137/1190966
AIRAPETIAN, R. A., Extending CR functions from piecewise smooth CR manifolds. Math. Sb., 134 (1987), 108–118 (Russian).
Aldo Andreotti and Mauro Nacinovich, Analytic convexity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 2, 287–372. MR 581145
Michèle Audin, Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagragiennes et plongements totalement réels, Comment. Math. Helv. 63 (1988), no. 4, 593–623 (French). MR 966952, DOI 10.1007/BF02566781
M. S. Baouendi and F. Trèves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2) 113 (1981), no. 2, 387–421. MR 607899, DOI 10.2307/2006990
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, Global convexity properties of some families of three-dimensional compact Levi-flat hypersurfaces, Trans. Amer. Math. Soc. 332 (1992), no. 1, 459–474. MR 1055805, DOI 10.1090/S0002-9947-1992-1055805-3
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
D. E. Barrett and J. E. Fornæss, On the smoothness of Levi-foliations, Publ. Mat. 32 (1988), no. 2, 171–177. MR 975896, DOI 10.5565/PUBLMAT_{3}2288_{0}5
Eric Bedford and Paolo De Bartolomeis, Levi flat hypersurfaces which are not holomorphically flat, Proc. Amer. Math. Soc. 81 (1981), no. 4, 575–578. MR 601733, DOI 10.1090/S0002-9939-1981-0601733-2
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
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, 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
Albert Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. MR 1211412
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
So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
CHIRKA, E. M., Analytic representation of CR functions. (Russian) Math. USSR Sbornik, 27 (1975), 526–553.
Michael Christ, Global $C^\infty$ irregularity of the $\overline \partial$-Neumann problem for worm domains, J. Amer. Math. Soc. 9 (1996), no. 4, 1171–1185. MR 1370592, DOI 10.1090/S0894-0347-96-00213-5
Mihnea Colţoiu, Complete locally pluripolar sets, J. Reine Angew. Math. 412 (1990), 108–112. MR 1074376, DOI 10.1515/crll.1990.412.108
John P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615–637. MR 657241, DOI 10.2307/2007015
Jean-Pierre Demailly, Cohomology of $q$-convex spaces in top degrees, Math. Z. 204 (1990), no. 2, 283–295. MR 1055992, DOI 10.1007/BF02570874
Klas Diederich and John Erik Fornaess, Pseudoconvex domains: an example with nontrivial Nebenhülle, Math. Ann. 225 (1977), no. 3, 275–292. MR 430315, DOI 10.1007/BF01425243
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
John Erik Fornaess and Alexander Nagel, The Mergelyan property for weakly pseudoconvex domains, Manuscripta Math. 22 (1977), no. 2, 199–208. MR 457779, DOI 10.1007/BF01167861
Franc Forstnerič, On totally real embeddings into $\textbf {C}^n$, Exposition. Math. 4 (1986), no. 3, 243–255. MR 880125
Franc Forstnerič, A contractible Levi-flat hypersurface which is a determining set for pluriharmonic functions, Ark. Mat. 44 (2006), no. 1, 87–91. MR 2237212, DOI 10.1007/s11512-005-0007-0
Étienne Ghys, L’invariant de Godbillon-Vey, Astérisque 177-178 (1989), Exp. No. 706, 155–181 (French). Séminaire Bourbaki, Vol. 1988/89. MR 1040572
Giuliana Gigante and Giuseppe Tomassini, Foliations with complex leaves, Differential Geom. Appl. 5 (1995), no. 1, 33–49. MR 1319934, DOI 10.1016/0926-2245(95)00004-N
Claude Godbillon, Dynamical systems on surfaces, Universitext, Springer-Verlag, Berlin-New York, 1983. Translated from the French by H. G. Helfenstein. MR 681119
Claude Godbillon, Feuilletages, Progress in Mathematics, vol. 98, Birkhäuser Verlag, Basel, 1991 (French). Études géométriques. [Geometric studies]; With a preface by G. Reeb. MR 1120547
Claude Godbillon and Jacques Vey, Un invariant des feuilletages de codimension $1$, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A92-A95 (French). MR 283816
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
André Haefliger, Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comment. Math. Helv. 32 (1958), 248–329 (French). MR 100269, DOI 10.1007/BF02564582
André Haefliger and Georges Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan, Enseign. Math. (2) 3 (1957), 107–125 (French). MR 89412
Gennadi Henkin and Jürgen Leiterer, Theory of functions on complex manifolds, Monographs in Mathematics, vol. 79, Birkhäuser Verlag, Basel, 1984. MR 774049
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
E. Kamke, Über die partielle Differentialgleichung, Math. Z. 41 (1936), no. 1, 56–66 (German). MR 1545604, DOI 10.1007/BF01180405
J. J. Kohn, A survey of the $\bar \partial$-Neumann problem, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 137–145. MR 740877, DOI 10.1090/pspum/041/740877
Christine Laurent-Thiébaut, Sur l’extension des fonctions CR dans une variété de Stein, Ann. Mat. Pura Appl. (4) 150 (1988), 141–151 (French, with English summary). MR 946033, DOI 10.1007/BF01761467
Christine Laurent-Thiébaut, Extension de formes différentielles CR, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 13, 539–542 (French, with English summary). MR 941620
Christine Laurent-Thiébaut, Sur l’équation de Cauchy-Riemann tangentielle dans une calotte strictement pseudoconvexe, Internat. J. Math. 16 (2005), no. 9, 1063–1079 (French, with English and French summaries). MR 2180065, DOI 10.1142/S0129167X05003223
Christine Laurent-Thiébaut and Jürgen Leiterer, On the Hartogs-Bochner extension phenomenon for differential forms, Math. Ann. 284 (1989), no. 1, 103–119. MR 995385, DOI 10.1007/BF01443508
Ch. Laurent-Thiébaut and E. Porten, Analytic extension from non-pseudoconvex boundaries and $A(D)$-convexity, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 3, 847–857 (English, with English and French summaries). MR 2008443
Guido Lupacciolu, Characterization of removable sets in strongly pseudoconvex boundaries, Ark. Mat. 32 (1994), no. 2, 455–473. MR 1318542, DOI 10.1007/BF02559581
Guido Lupacciolu and Giuseppe Tomassini, An extension theorem for CR-functions, Ann. Mat. Pura Appl. (4) 137 (1984), 257–263 (Italian, with English summary). MR 772261, DOI 10.1007/BF01789398
Xosé M. Masa, Alexander-Spanier cohomology of foliated manifolds, Illinois J. Math. 46 (2002), no. 4, 979–998. MR 1988246
S. Yu. Nemirovskiĭ, Stein domains with Levi-plane boundaries on compact complex surfaces, Mat. Zametki 66 (1999), no. 4, 632–635 (Russian); English transl., Math. Notes 66 (1999), no. 3-4, 522–525 (2000). MR 1747093, DOI 10.1007/BF02679105
S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248–278 (Russian). MR 0200938
Takeo Ohsawa, On the Levi-flats in complex tori of dimension two, Publ. Res. Inst. Math. Sci. 42 (2006), no. 2, 361–377. MR 2250063
Hugo Rossi, Holomorphically convex sets in several complex variables, Ann. of Math. (2) 74 (1961), 470–493. MR 133479, DOI 10.2307/1970292
Richard Sacksteder, Foliations and pseudogroups, Amer. J. Math. 87 (1965), 79–102. MR 174061, DOI 10.2307/2373226
Mei-Chi Shaw, $L^p$ estimates for local solutions of $\overline \partial _\textrm {b}$ on strongly pseudo-convex CR manifolds, Math. Ann. 288 (1990), no. 1, 35–62. MR 1070923, DOI 10.1007/BF01444520
Mei-Chi Shaw, Local existence theorems with estimates for $\overline \partial _b$ on weakly pseudo-convex CR manifolds, Math. Ann. 294 (1992), no. 4, 677–700. MR 1190451, DOI 10.1007/BF01934348
N. V. Shcherbina, Decomposition of a common boundary of two domains of holomorphy into analytic curves, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1106–1123, 1136 (Russian). MR 675533
Yum Tong Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976/77), no. 1, 89–100. MR 435447, DOI 10.1007/BF01390170
Yum-Tong Siu, $\overline \partial$-regularity for weakly pseudoconvex domains in compact Hermitian symmetric spaces with respect to invariant metrics, Ann. of Math. (2) 156 (2002), no. 2, 595–621. MR 1933078, DOI 10.2307/3597199
Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363–386. MR 626955, DOI 10.1007/BF01679703
STRAUBE, E. J., A sufficient condition for global regularity of the $\overline {\partial }$-Neumann operator. Preprint, 2005. [arXiv:math.CV/0510354]
—, Aspects of the $L^2$-Sobolev theory of the $\overline {\partial }$-Neumann problem. Proceedings of the International Congress of Mathematicians, Madrid, 2006.
Emil J. Straube and Marcel K. Sucheston, Plurisubharmonic defining functions, good vector fields, and exactness of a certain one form, Monatsh. Math. 136 (2002), no. 3, 249–258. MR 1919648, DOI 10.1007/s006050200044
Emil J. Straube and Marcel K. Sucheston, Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with $\overline \partial$, Trans. Amer. Math. Soc. 355 (2003), no. 1, 143–154. MR 1928081, DOI 10.1090/S0002-9947-02-03133-1
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
Philippe Tondeur, Geometry of foliations, Monographs in Mathematics, vol. 90, Birkhäuser Verlag, Basel, 1997. MR 1456994, DOI 10.1007/978-3-0348-8914-8
WAZEWSKY, T., Sur l’équation $Pp+Qq=0$. Mathematica, 8 (1934), 103–116; 9 (1935), 179–182.