Compactness of the dbar-Neumann problem and Stein neighborhood bases

Abstract

This dissertation consists of two parts. In the first part we show that for 1 k 1, a complex manifold M of dimension at least k in the boundary of a smooth bounded pseudoconvex domain in Cn is an obstruction to compactness of the @- Neumann operator on (p, q)-forms for 0 p k n, provided that at some point of M, the Levi form of b has the maximal possible rank n − 1 − dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction to compactness of the @-Neumann operator on (p, 1)-forms, provided that at some point of the disc, the Levi form has only one vanishing eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one vanishing eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. In the second part we obtain a weaker and quantified version of McNeal’s Property ( eP) which still implies the existence of a Stein neighborhood basis. Then we give some applications on domains in C2 with a defining function that is plurisubharmonic on the boundary.