Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Spectrum of the complex Laplacian on product domains

Author(s): Debraj Chakrabarti
Journal: Proc. Amer. Math. Soc. 138 (2010), 3187-3202.
MSC (2010): Primary 32W05; Secondary 35P10, 35N15
Posted: May 12, 2010
MathSciNet review: 2653944
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We show that the spectrum of the complex Laplacian $\Box$ on a product of Hermitian manifolds is the Minkowski sum of the spectra of the complex Laplacians on the factors. We use this to show that the range of the Cauchy-Riemann operator $\overline{\partial}$ is closed on a product manifold, provided it is closed on each factor manifold.

References:

1.: Bertrams, Julia: Randregularität von Lösungen der $\overline\partial$ -Gleichung auf dem Polyzylinder und zweidimensionalen analytischen Polyedern. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1986. MR 867085 (88b:32044)
2.: Brüning, J. and Lesch, M.: Hilbert complexes. J. Funct. Anal. 108 (1992), no. 1, 88-132. MR 1174159 (93k:58208)
3.: Chakrabarti, Debraj and Shaw, Mei-Chi: The Cauchy-Riemann equations on a product domain. Preprint. Available at arXiv:0911.0103
4.: Cheeger, Jeff: On the Hodge theory of Riemannian pseudomanifolds. Proc. Sympos. Pure Math. 36, Amer. Math. Soc, Providence, RI, 1980, 91-145. MR 573430 (83a:58081)
5.: Cheeger, Jeff: Spectral geometry of singular Riemannian spaces. J. Differential Geom. 18, no. 4 (1983), 575-657. MR 730920 (85d:58083)
6.: Chen, So-Chin and Shaw, Mei-Chi: Partial differential equations in several complex variables. AMS/IP Studies in Advanced Mathematics, 19. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297 (2001m:32071)
7.: de Rham, Georges: Variétés différentiables. Formes, courants, formes harmoniques. Hermann et Cie, Paris, 1955. MR 0068889 (16:957b)
8.: Ehsani, Dariush: Solution of the $\overline\partial$ -Neumann problem on a non-smooth domain. Indiana Univ. Math. J. 52 (2003), no. 3, 629-666. MR 1986891 (2004f:32051)
9.: Ehsani, Dariush: Solution of the $\overline\partial$ -Neumann problem on a bi-disc. Math. Res. Lett. 10 (2003), no. 4, 523-533. MR 1995791 (2004f:32052)
10.: Ehsani, Dariush: The $\overline\partial$ -Neumann problem on product domains in $\mathbb{C}^n$ . Math. Ann. 337 (2007), 797-816. MR 2285738 (2007j:32041)
11.: Folland, Gerald B. and Kohn, Joseph J.:
The Neumann problem for the Cauchy-Riemann complex. Annals of Mathematics Studies, No. 75. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1972. MR 0461588 (57:1573)
12.: Fu, Siqi: Spectrum of the $\overline\partial$ -Neumann Laplacian on polydiscs. Proc. Amer. Math. Soc. 135 (2007), no. 3, 725-730. MR 2262868 (2007h:32056)
13.: Halmos, P. R: What does the spectral theorem say? Amer. Math. Monthly 70 (1963), 241-247. MR 0150600 (27:595)
14.: Reed, Michael and Simon, Barry: Methods of modern mathematical physics, Vol. I: Functional analysis. Second edition. Academic Press, Inc., New York, 1980. MR 751959 (85e:46002)
15.: Shaw, Mei-Chi: Global solvability and regularity for $\overline{\partial}$ on an annulus between two weakly pseudoconvex domains. Trans. Amer. Math. Soc. 291 (1985), 255-267. MR 797058 (86m:32030)
16.: Shaw, Mei-Chi: The closed range property for $\overline\partial$ , on domains with pseudoconcave boundary. Proceedings of the Complex Analysis Conference, Fribourg, Switzerland (2008).
17.: Teschl, Gerald: Mathematical methods in quantum mechanics. With applications to Schrödinger operators. Graduate Studies in Mathematics, 99, American Mathematical Society, Providence, RI, 2009. MR 2499016
18.: Zucker, Steven: $L_{2}$ cohomology of warped products and arithmetic groups. Invent. Math. 70 (1982), no. 2, 169-218. MR 684171 (86j:32063)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|