Embeddability of some strongly pseudoconvex CR manifolds
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- by George Marinescu and Nader Yeganefar PDF
- Trans. Amer. Math. Soc. 359 (2007), 4757-4771 Request permission
Abstract:
We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are boundaries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kähler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.References
- Aldo Andreotti, Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves, Bull. Soc. Math. France 91 (1963), 1–38 (French). MR 152674, DOI 10.24033/bsmf.1587
- Nicolae Anghel, An abstract index theorem on noncompact Riemannian manifolds, Houston J. Math. 19 (1993), no. 2, 223–237. MR 1225459
- Aldo Andreotti and Yum-tong Siu, Projective embedding of pseudoconcave spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 24 (1970), 231–278. MR 265633
- Aldo Andreotti and Giuseppe Tomassini, Some remarks on pseudoconcave manifolds, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 85–104. MR 0265632
- Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
- Florin Alexandru Belgun, Normal CR structures on compact 3-manifolds, Math. Z. 238 (2001), no. 3, 441–460. MR 1869692, DOI 10.1007/s002090100260
- Olivier Biquard and Marc Herzlich, A Burns-Epstein invariant for ACHE 4-manifolds, Duke Math. J. 126 (2005), no. 1, 53–100. MR 2110628, DOI 10.1215/S0012-7094-04-12612-0
- John Bland and C. L. Epstein, Embeddable CR-structures and deformations of pseudoconvex surfaces. I. Formal deformations, J. Algebraic Geom. 5 (1996), no. 2, 277–368. MR 1374711
- 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, Exp. No. 9, Centre Math., École Polytech., Paris, 1975, pp. 14 (French). MR 0409893
- Daniel M. Burns and Charles L. Epstein, Embeddability for three-dimensional CR-manifolds, J. Amer. Math. Soc. 3 (1990), no. 4, 809–841. MR 1071115, DOI 10.1090/S0894-0347-1990-1071115-4
- Daniel M. Burns Jr., Global behavior of some tangential Cauchy-Riemann equations, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 51–56. MR 535588
- David Catlin, A Newlander-Nirenberg theorem for manifolds with boundary, Michigan Math. J. 35 (1988), no. 2, 233–240. MR 959270, DOI 10.1307/mmj/1029003750
- J.-P. Demailly, Complex analytic and differential geometry, published online at www-fourier.ujf-grenoble.fr/˜demailly/lectures.html, 2001.
- Patrick Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. (2) 111 (1980), no. 3, 435–476. MR 577132, DOI 10.2307/1971104
- P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. MR 336648, DOI 10.2140/pjm.1973.46.45
- Charles L. Epstein and Gennadi M. Henkin, Stability of embeddings for pseudoconcave surfaces and their boundaries, Acta Math. 185 (2000), no. 2, 161–237. MR 1819994, DOI 10.1007/BF02392810
- Elisha Falbel, Nonembeddable CR-manifolds and surface singularities, Invent. Math. 108 (1992), no. 1, 49–65. MR 1156386, DOI 10.1007/BF02100599
- Hansjörg Geiges, Normal contact structures on $3$-manifolds, Tohoku Math. J. (2) 49 (1997), no. 3, 415–422. MR 1464186, DOI 10.2748/tmj/1178225112
- Hans Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368 (German). MR 137127, DOI 10.1007/BF01441136
- R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983, DOI 10.1007/BFb0063413
- M. Gromov, Kähler hyperbolicity and $L_2$-Hodge theory, J. Differential Geom. 33 (1991), no. 1, 263–292. MR 1085144, DOI 10.4310/jdg/1214446039
- F. Reese Harvey and H. Blaine Lawson Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975), no. 2, 223–290. MR 425173, DOI 10.2307/1971032
- Ernst Heintze and Hans-Christoph Im Hof, Geometry of horospheres, J. Differential Geometry 12 (1977), no. 4, 481–491 (1978). MR 512919
- Dieter Heunemann, Extension of the complex structure from Stein manifolds with strictly pseudoconvex boundary, Math. Nachr. 128 (1986), 57–64. MR 855943, DOI 10.1002/mana.19861280105
- J. J. Kohn and Hugo Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451–472. MR 177135, DOI 10.2307/1970624
- László Lempert, Algebraic approximations in analytic geometry, Invent. Math. 121 (1995), no. 2, 335–353. MR 1346210, DOI 10.1007/BF01884302
- George Marinescu and Tien-Cuong Dinh, On the compactification of hyperconcave ends, C. R. Math. Acad. Sci. Paris 342 (2006), no. 9, 675–680 (English, with English and French summaries). MR 2225875, DOI 10.1016/j.crma.2006.02.038
- Alan Nadel and Hajime Tsuji, Compactification of complete Kähler manifolds of negative Ricci curvature, J. Differential Geom. 28 (1988), no. 3, 503–512. MR 965227
- Terrence Napier and Mohan Ramachandran, The $L^2\ \overline \partial$-method, weak Lefschetz theorems, and the topology of Kähler manifolds, J. Amer. Math. Soc. 11 (1998), no. 2, 375–396. MR 1477601, DOI 10.1090/S0894-0347-98-00257-4
- Raghavan Narasimhan, Imbedding of holomorphically complete complex spaces, Amer. J. Math. 82 (1960), 917–934. MR 148942, DOI 10.2307/2372949
- Takeo Ohsawa, Holomorphic embedding of compact s.p.c. manifolds into complex manifolds as real hypersurfaces, Differential geometry of submanifolds (Kyoto, 1984) Lecture Notes in Math., vol. 1090, Springer, Berlin, 1984, pp. 64–76. MR 775145, DOI 10.1007/BFb0101567
- H. Rossi, Attaching analytic spaces to an analytic space along a pseudoconcave boundary, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 242–256. MR 0176106
- Yum Tong Siu and Shing Tung Yau, Compactification of negatively curved complete Kähler manifolds of finite volume, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 363–380. MR 645748
- Noboru Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9, Kinokuniya Book Store Co., Ltd., Tokyo, 1975. MR 0399517
- Hajime Urakawa, Variational problems over strongly pseudoconvex CR manifolds, Differential geometry (Shanghai, 1991) World Sci. Publ., River Edge, NJ, 1993, pp. 233–242. MR 1341616
- S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), no. 1, 25–41. MR 520599, DOI 10.4310/jdg/1214434345
- N. Yeganefar, $L^2$-cohomology of negatively curved Kähler manifolds of finite volume, to appear in Geom. and Funct. Analysis, arXiv:math.DG/0402056, 2004.
Additional Information
- George Marinescu
- Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
- Address at time of publication: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D50931 Köln, Germany
- MR Author ID: 819533
- Email: george@mathematik.hu-berlin.de, gmarines@math.uni-koeln.de
- Nader Yeganefar
- Affiliation: Département de Mathématiques, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes cedex 03, France
- Address at time of publication: CMI, Université de Provence, 39 Rue Frédéric Joliot Curie, 13453 Marseille cedex 13, France
- Email: nader.yeganefar@math.univ-nantes.fr, Nader.Yeganefar@cmi.univ-mrs.fr
- Received by editor(s): January 10, 2005
- Received by editor(s) in revised form: April 18, 2005
- Published electronically: April 24, 2007
- Additional Notes: The second-named author was (partially) supported by the European Commission through the Research Training Network HPRN-CT-1999-00118 “Geometric Analysis”.
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 4757-4771
- MSC (2000): Primary 32V30, 32V15, 32Q05
- DOI: https://doi.org/10.1090/S0002-9947-07-04047-0
- MathSciNet review: 2320650