The G-Fredholm Property of the \bar\partial-Neumann Problem

TL;DR

This paper demonstrates that the complex Laplacian on a strongly pseudoconvex manifold with a group action exhibits G-Fredholm properties, including finite G-dimension of kernels and finite codimension of images, extending classical Fredholm theory.

Contribution

It establishes the G-Fredholm property of the complex Laplacian and boundary Laplacian on certain pseudoconvex manifolds with group actions, a novel extension of Fredholm theory.

Findings

Kernel of  with q>0 is G-invariant and finite G-dimension.

Image of  contains a G-invariant subspace of finite codimension.

 is a G-Fredholm operator under the given conditions.

Abstract

Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ be the total space of a principal bundle $G\to M\to X$ 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 complex Laplacian $\square$ on $M$ has the following properties: The kernel of $\square$ restricted to the forms $\Lambda^{p,q}$ with $q$ positive is a closed, $G$-invariant subspace in $L^{2}(M,\Lambda^{p,q})$ of finite $G$-dimension. Secondly, we show that if $q$ is positive, then the image of $\square$ contains a closed, $G$-invariant subspace of finite codimension in $L^{2}(M,\Lambda^{p,q})$. These two properties taken together amount to saying that $\square$ is a $G$-Fredholm operator. The boundary Laplacian has similar properties.