Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Stein fillability and the realization of contact manifolds

Authors: C. Denson Hill and Mauro Nacinovich
Journal: Proc. Amer. Math. Soc. 133 (2005), 1843-1850
MSC (2000): Primary 53D10, 32V15, 35N99
Published electronically: January 21, 2005
MathSciNet review: 2120286
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though a border always exists, its germ is not unique; nevertheless the germ of the Dolbeault cohomology of any border is unique. We also point out that any Stein fillable compact contact $3$-manifold has a geometric realization in $\mathbb{C} ^{4}$ via an embedding, or in $\mathbb{C} ^{3}$ via an immersion.

References [Enhancements On Off] (What's this?)

  • [AF] A.Andreotti, T.Frankel, The Lefschetz theorem on hyperplane sections, Ann. Math. 69 (1959), 713-717. MR 0177422 (31:1685)
  • [AG] A.Andreotti, H.Grauert, Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193-259. MR 0150342 (27:343)
  • [AH1] A.Andreotti, C.D.Hill, E. E. Levi convexity and the Hans Lewy problem. I. Reduction to vanishing theorems, Ann. Scuola Norm. Sup. Pisa 26 (1972), 325-363. MR 0460725 (57:718)
  • [AH2] A.Andreotti, C.D.Hill, E. E. Levi convexity and the Hans Lewy problem. II. Vanishing theorems, Ann. Scuola Norm. Sup.Pisa 26 (1972), 747-806. MR 0477150 (57:16693)
  • [Bi] E. Bishop, Mappings of partially analytic spaces, Amer. J. Math. 83 (1961), 209-242. MR 0123732 (23:A1054)
  • [BM] L.Boutet de Monvel, Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz 1974-1975; Équations aux derivées partielles linéaires et non linéaires, vol. Exp. No. 9, Centre Math. École Polytech., Paris, 1975, pp. 1-14.
  • [Ca] D.Catlin, A Newlander-Nirenberg theorem for manifolds with boundary, Michigan Math. J. 35 (1988), 233-240. MR 0959270 (89j:32026)
  • [Ch] S. Cho, Extension of complex structures on weakly pseudoconvex compact complex manifolds with boundary, Math. Z. 211 (1992), 105-120. MR 1179783 (94h:32030)
  • [DA] J. D'Angelo, Finite type conditions and subelliptic estimates, Modern methods in complex analysis (1992), Ann. of Math. Stud., 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 63-78. MR 1369134 (96k:32030)
  • [El1] Y.M.Eliashberg, Filling by holomorphic discs and its applications, Geometry of low-dimensional manifolds, 2, London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45-47. MR 1171908 (93g:53060)
  • [El2] Y.M.Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990), 29-46. MR 1044658 (91k:32012)
  • [El3] Y.M.Eliashberg, Symplectic topology in the nineties, Differential Geom. Appl. 9 (1998), 59-88. MR 1636301 (2000a:53136)
  • [Go] R.E.Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998), 619-693. MR 1668563 (2000a:57070)
  • [He] D. Heunemann, Extension of the complex structure from Stein manifolds with strictly pseudoconvex boundary, Math. Nachr. 128 (1986), 87-101. MR 0855943 (87j:32057)
  • [H1] C.D.Hill, What is the notion of a complex manifold with a smooth boundary?, Algebraic Analysis I (Kashiwara and Kawai, eds.), Academic Press, Boston, MA, 1988, pp. 185-201. MR 0992454 (90e:32009)
  • [H2] C.D.Hill, A family of exotic $CR$ structures on $S\sp{3}\times S\sp{2}$, Analyse complexe multivariable: récents développements (Guadeloupe, 1988) Sem. Conf., 5, EditEl, Rende, 1991, pp. 105-110. MR 1228874 (94f:32039)
  • [H3] C.D.Hill, Counterexamples to Newlander-Nirenberg up to the boundary, Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., 52, Part 3, Amer. Math. Soc.,, Providence, RI, 1991, pp. 191-197. MR 1128593 (92m:32027)
  • [HJ] N.Hanges, H.Jacobowitz, A remark on almost complex structures with boundary, Amer. J. Math. 111 (1989), 53-64. MR 0980299 (90m:32038)
  • [HN1] C.D.Hill, M.Nacinovich, A collar neighborhood theorem for a complex manifold, Rend. Sem. Mat. Univ. Padova 91 (1994), 24-30. MR 1289628 (95g:32015)
  • [HN2] C.D.Hill, M.Nacinovich, The topology of Stein $CR$ manifolds and the Lefschetz theorem, Ann. Inst. Fourier (Grenoble) 43 (1993), 459-468. MR 1220278 (94d:32012)
  • [HN3] C.D.Hill, M.Nacinovich, Pseudoconcave $CR$ manifolds, Complex analysis and geometry (Trento, 1993), Lecture Notes in Pure and Appl. Math., 173, Dekker, New York, 1996, pp. 275-297. MR 1365978 (97c:32013)
  • [La] H.B. Laufer, On the infinite dimensionality of the Dolbeault cohomology groups, Proc. Amer. Math. Soc. 52 (1975), 293-296. MR 0379887 (52:792)
  • [Le1] L. Lempert, On three-dimensional Cauchy-Riemann manifolds, J. AMS 5 (1992), 923-969. MR 1157290 (94d:32025)
  • [Le2] L.Lempert, Embeddings of three-dimensional Cauchy-Riemann manifolds, Math. Ann. 300 (1994), 1-15. MR 1289827 (95j:32029)
  • [Le3] L. Lempert, Algebraic approximations in analytic geometry, Invent. Math. 121 (1995), 335-353. MR 1346210 (97k:32021)
  • [LiM] P.Lisca, G.Matic, G. Stein $4$-manifolds with boundary and contact structures.; Symplectic, contact and low-dimensional topology (Athens, GA, 1996), Topology Appl. 88 (1998), 55-66. MR 1634563 (99f:57037)
  • [Na] R.Narasimhan, Imbedding of holomorphically complete complex spaces, Amer. J. Math. 82 (1960), 917-934. MR 0148942 (26:6438)
  • [Oh] T. Ohsawa, Holomorphic embeddings of compact s.p.c. manifolds into complex manifolds as real hypersurfaces, Differential geometry of submanifolds (Kyoto, 1984), Lecture Notes in Math., 1090, Springer, Berlin, 1984, pp. 64-76. MR 0775145 (86j:32047)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53D10, 32V15, 35N99

Retrieve articles in all journals with MSC (2000): 53D10, 32V15, 35N99

Additional Information

C. Denson Hill
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794

Mauro Nacinovich
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 1 - 00133 - Roma, Italy

Keywords: Stein manifold, contact manifold
Received by editor(s): November 19, 2003
Received by editor(s) in revised form: March 2, 2004
Published electronically: January 21, 2005
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society