Perez, Joe J. The \(G\)-Fredholm property of the \(\bar{\partial}\)-Neumann problem. (English) Zbl 1187.32032 J. Geom. Anal. 19, No. 1, 87-106 (2009). Let \(M\) be a complex manifold with boundary which is strongly pseudoconvex. Let \(G\) be a unimodular Lie group acting freely by holomorphic transformations on \(M\) so that \(M/G\) is compact. It is shown that, for \(q>0,\) the Kohn Laplacian \(\square = \overline\partial^* \, \overline \partial + \overline \partial \, \overline \partial^*\) in \(L^2(M, \Lambda^{p,q})\) is \(G\)- Fredholm, which means that \({\text{dim}}_M {\text{ker}}(\square ) < \infty.\) As a corollary one obtains that the reduced \(L^2\)- Dolbeault cohomology spaces \(L^2 \tilde H^{p,q} (M)\) of \(M\) are finite \(G\)- dimensional for \(q>0.\) The boundary Laplacian \(\square_b\) has similar properties. Reviewer: Fritz Haslinger (Wien) Cited in 6 Documents MSC: 32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators 35H20 Subelliptic equations 46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.) Keywords:\(\overline \partial\)-Neumann problem; subelliptic operators PDFBibTeX XMLCite \textit{J. J. Perez}, J. Geom. Anal. 19, No. 1, 87--106 (2009; Zbl 1187.32032) Full Text: DOI arXiv References: [1] Atiyah, M.F.: Elliptic operators, discrete groups, and von Neumann algebras. Soc. Math. Fr. Astérisque 32-3, 43–72 (1976) · Zbl 0323.58015 [2] Breuer, M.: Fredholm theories in von Neumann algebras, I, II. Math. Ann. 178, 243–254 (1968) · Zbl 0162.18701 · doi:10.1007/BF01350663 [3] Breuer, M.: Fredholm theories in von Neumann algebras, I, II. Math. Ann. 180, 313–325 (1969) · Zbl 0175.44102 · doi:10.1007/BF01351884 [4] Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Grundlehren der Mathematischen Wissenschaften, vol. 298. Springer, Berlin (1992) · Zbl 0744.58001 [5] Brudnyi, A.: On holomorphic L 2 functions on coverings of strongly pseudoconvex manifolds. arXiv.org/pdf/math.CV/0508237 · Zbl 1165.32014 [6] Coburn, L.A., Moyer, R.D., Singer, I.M.: C *-algebras of almost-periodic pseudo-differential operators. Acta Math. 130, 279–307 (1973) · Zbl 0263.47042 · doi:10.1007/BF02392269 [7] Connes, A., Moscovici, H.: The L 2-index theorem for homogeneous spaces of Lie groups. Ann. Math. 115(2), 291–330 (1982) · Zbl 0515.58031 · doi:10.2307/1971393 [8] Engliš, M.: Pseudolocal estimates for \(\bar{\partial}\) on general pseudoconvex domains. Indiana Univ. Math. J. 50(4), 1593–1607 (2001). Erratum. Indiana Univ. Math. J. (to appear) · Zbl 1044.32029 · doi:10.1512/iumj.2001.50.2085 [9] Folland, G.B., Kohn, J.J.: The Neumann Problem for the Cauchy-Riemann Complex. Ann. Math. Studies, vol. 75. Princeton University Press, Princeton (1972) · Zbl 0247.35093 [10] Fedosov, B., Shubin, M.A.: The index of random elliptic operators, I, II. Mat. Sb. (N.S.) 106(148)(1), 108–140, 144 (1978) · Zbl 0448.47033 [11] Fedosov, B., Shubin, M.A.: The index of random elliptic operators, I, II. Mat. Sb. (N.S.) 106(148)(3), 455–483, 496 (1978) · Zbl 0448.47033 [12] Gromov, M., Henkin, G., Shubin, M.: Holomorphic L 2 functions on coverings of pseudoconvex manifolds. Geom. Funct. Anal. 8(3), 552–585 (1998) · Zbl 0926.32011 · doi:10.1007/s000390050066 [13] Gromov, M.: Curvature, diameter, and Betti numbers. Comment. Math. Helv. 56(2), 179–195 (1981) · Zbl 0467.53021 · doi:10.1007/BF02566208 [14] Kohn, J.J.: Harmonic integrals on strongly pseudoconvex manifolds, I, II. Ann. Math. 78, 112–148 (1963) · Zbl 0161.09302 · doi:10.2307/1970506 [15] Kohn, J.J.: Harmonic integrals on strongly pseudoconvex manifolds, I, II. Ann. Math. 79, 450–472 (1964) · Zbl 0178.11305 · doi:10.2307/1970404 [16] Lions, J.L., Magenes, E.: Non-Homgeneous Boundary Value Problems and Applications. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 181. Springer, Berlin (1972) · Zbl 0223.35039 [17] Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 17. Springer, Berlin (1991) · Zbl 0732.22008 [18] Morrey, C.B.: The analytic embedding of abstract real-analytic manifolds. Ann. Math. 68, 159–201 (1968) · Zbl 0090.38401 · doi:10.2307/1970048 [19] Pedersen, G.K.: C*-Algebras and their Automorphism Groups. London Mathematical Society Monographs, vol. 14. Academic Press, San Diego (1979) · Zbl 0416.46043 [20] Shubin, M.A.: L 2 Riemann-Roch theorem for elliptic operators. Geom. Funct. Anal. 5(2), 482–527 (1995) · Zbl 0840.58045 · doi:10.1007/BF01895677 [21] Shubin, M.A.: Spectral theory of elliptic operators on noncompact manifolds. Astérisque 207(5), 35–108 (1992). Méthodes semi-classiques, vol. 1 (Nantes, 1991) [22] Takesaki, M.: Theory of Operator Algebras, vol. I. Springer, Berlin (1979) · Zbl 0436.46043 [23] Todor, R., Chiose, I., Marinescu, G.: L 2-Riemann-Roch inequalities for covering manifolds. arXiv.org/pdf/math.AG/0002049 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.