Global regularity for the $\bar{\partial }$-Neumann problem on pseudoconvex manifolds

Abstract

We establish general sufficient conditions for exact (and global) regularity in the $\bar{\partial }$-Neumann problem on $(p,q)$-forms, $0\le p\le n$ and $1\le q\le n$, on a pseudoconvex domain $\hat{I}\text{c◯}$ with smooth boundary $b\hat{I}\text{c◯}$ in an $n$-dimensional complex manifold $M$. Our hypotheses include two assumptions: 1) $M$ admits a function that is strictly plurisubharmonic acting on $(p_0,q_0)$-forms in a neighborhood of $b\hat{I}\text{c◯}$ for some fixed $0\le p_0\le n$, $1\le q_0\le n$, or $M$ is a Kähler metric whose holomorphic bisectional curvature acting $(p,q)$-forms is positive; and 2) there exists a family of vector fields $T_{\hat{I}}\mu$ that are transverse to the boundary $b\hat{I}\text{c◯}$ and generate one forms, which when applied to $(p,q)$-forms, $0\le p\le n$ and $q_0\le q\le n$, satisfy a "weak form" of the compactness estimate. We also provide examples and applications of our main theorems.

28 pages. Comments welcome!