Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G→M→X 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
on M has the following properties: The kernel of □ restricted to the forms Λp,q with q>0 is a closed, G-invariant subspace in L2(M,Λp,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 L2(M,Λp,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 L2-Dolbeault cohomology spaces
of M are finite G-dimensional for q>0. The boundary Laplacian □b has similar properties.
-Neumann problem Subelliptic operators
Mathematics Subject Classification (2000)
32W05 35H20 46L99
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