The Cauchy-Riemann equations are fundamental in one and several complex variables. Holomorphic functions, for example, satisfy the homogeneous Cauchy-Riemann equations. In complex Euclidean space of dimension $n\ge 2$, the necessary and sufficient condition for the solvability of the Cauchy-Riemann equations is that the domain is pseudoconvex.
In this talk we relate pseudoconvexity with the vanishing of $L^2$ Dolbeault cohomology groups. On a pseudoconvex domain in a complex manifold, the $L^2$ Dolbeault cohomology might not even be Hausdorff. Recent results on the
$L^2$ closed range property for $\bar\partial$ on an annulus between two pseudoconvex domains will be discussed. One can even characterize such domains through the spectral theory of their $L^2$ Dolbeault cohomology groups, thus hearing pseudoconvexity of the boundary. (Joint work with Debraj Chakrabarti, Siqi Fu, and Christine Laurent-Thi\'ebaut). |