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The G-Fredholm Property of the \(\bar{\partial}\) -Neumann Problem

  • Joe J. PerezEmail author
Article

Abstract

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle GMX so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M, then the Laplacian \(\square=\bar{\partial}^{*}\bar{\partial}+\bar{\partial}\bar{\partial}^{*}\) on M has the following properties: The kernel of restricted to the forms Λ p,q with q>0 is a closed, G-invariant subspace in L 2(M p,q ) of finite G-dimension. Secondly, we show that if q>0, then the image of contains a closed, G-invariant subspace of finite G-codimension in L 2(M p,q ). These two properties taken together amount to saying that is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L 2-Dolbeault cohomology spaces \(L^{2}\bar{H}^{p,q}(M)\) of M are finite G-dimensional for q>0. The boundary Laplacian b has similar properties.

Keywords

\(\bar{\partial}\) -Neumann problem Subelliptic operators 

Mathematics Subject Classification (2000)

32W05 35H20 46L99 

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Copyright information

© Mathematica Josephina, Inc. 2008

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityKingsvilleUSA

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