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The Serre duality theorem for a non-compact weighted CR manifold. (English) Zbl 1219.32018

Summary: 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.

MSC:

32V20 Analysis on CR manifolds
53C17 Sub-Riemannian geometry
58A14 Hodge theory in global analysis
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[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). · Zbl 0851.31008 · doi:10.1007/BF01759352
[2] Albert Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. · Zbl 0760.32001
[3] David E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976. · Zbl 0319.53026
[4] Wei-Liang Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98 – 105 (German). · Zbl 0022.02304 · doi:10.1007/BF01450011
[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. · Zbl 0247.35093
[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. · Zbl 0848.32003
[7] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147 – 171. · Zbl 0156.10701 · doi:10.1007/BF02392081
[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. · Zbl 1145.32019
[9] Alexander Isaev, Lectures on the automorphism groups of Kobayashi-hyperbolic manifolds, Lecture Notes in Mathematics, vol. 1902, Springer, Berlin, 2007. · Zbl 1196.32017
[10] J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. of Math. (2) 78 (1963), 112 – 148. , https://doi.org/10.2307/1970506 J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. II, Ann. of Math. (2) 79 (1964), 450 – 472. · Zbl 0178.11305 · doi:10.2307/1970404
[11] J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443 – 492. · Zbl 0125.33302 · doi:10.1002/cpa.3160180305
[12] Jun Masamune, Essential self-adjointness of a sublaplacian via heat equation, Comm. Partial Differential Equations 30 (2005), no. 10-12, 1595 – 1609. · Zbl 1084.53028 · doi: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. · Zbl 1164.32006
[14] Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. 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. Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0242.46001
[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. · Zbl 0331.53025
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