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)

 

 

The Serre duality theorem for a non-compact weighted CR manifold


Authors: Mitsuhiro Itoh, Jun Masamune and Takanari Saotome
Journal: Proc. Amer. Math. Soc. 136 (2008), 3539-3548
MSC (2000): Primary 32V20, 53C17; Secondary 58A14, 14F15
Published electronically: June 11, 2008
MathSciNet review: 2415038
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that the Hodge decomposition and Serre duality hold on a non-compact weighted CR manifold with negligible boundary. A complete CR manifold has negligible boundary. Some examples of complete CR manifolds are presented.


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

  • 1. M. Biroli and U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. (4) 169 (1995), 125–181 (English, with English and Italian summaries). MR 1378473, 10.1007/BF01759352
  • 2. Albert Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. MR 1211412
  • 3. David E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976. MR 0467588
  • 4. Wei-Liang Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98–105 (German). MR 0001880
  • 5. G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 75. MR 0461588
  • 6. C. Denson Hill and M. Nacinovich, Duality and distribution cohomology of CR manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 2, 315–339. MR 1354910
  • 7. Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 0222474
  • 8. Mitsuhiro Itoh and Takanari Saotome, The Serre duality theorem for holomorphic vector bundles over a strongly pseudo-convex manifold, Tsukuba J. Math. 31 (2007), no. 1, 197–204. MR 2337126
  • 9. Alexander Isaev, Lectures on the automorphism groups of Kobayashi-hyperbolic manifolds, Lecture Notes in Mathematics, vol. 1902, Springer, Berlin, 2007. MR 2352328
  • 10. J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. of Math. (2) 78 (1963), 112–148. MR 0153030
    J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. II, Ann. of Math. (2) 79 (1964), 450–472. MR 0208200
  • 11. J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. MR 0181815
  • 12. Jun Masamune, Essential self-adjointness of a sublaplacian via heat equation, Comm. Partial Differential Equations 30 (2005), no. 10-12, 1595–1609. MR 2182306, 10.1080/03605300500299935
  • 13. J. Masamune, Vanishing and conservativeness of harmonic forms of a non-compact CR manifold, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), no. 2, 79-102.
  • 14. Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
  • 15. Noboru Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya Book-Store Co., Ltd., Tokyo, 1975. Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9. MR 0399517

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32V20, 53C17, 58A14, 14F15

Retrieve articles in all journals with MSC (2000): 32V20, 53C17, 58A14, 14F15


Additional Information

Mitsuhiro Itoh
Affiliation: Institute of Mathematics, University of Tsukuba, 305-8751, Tsukuba, Japan
Email: itohm@sakura.cc.tsukuba.ac.jp

Jun Masamune
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609-2280
Email: masamune@wpi.edu

Takanari Saotome
Affiliation: Graduate School of Pure and Applied Sciences, University of Tsukuba, 305-8571, Tsukuba, Japan
Email: tsaotome@math.tsukuba.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-08-09498-7
Keywords: Strongly pseudo-convex manifold, CR manifold, Serre duality, Hodge decomposition, Witten-Kohn Laplacian, weighted Laplacian
Received by editor(s): July 17, 2007
Published electronically: June 11, 2008
Communicated by: Mikhail Shubin
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.