Global regularity for the ∂-Neumann problem on pseudoconvex manifolds
Abstract
We establish general sufficient conditions for exact (and global) regularity in the ∂-Neumann problem on (p,q)-forms, 0 ≤ p ≤ n and 1≤ q ≤ n, on a pseudoconvex domain with smooth boundary b in an n-dimensional complex manifold M. Our hypotheses include two assumptions: 1) M admits a function that is strictly plurisubharmonic acting on (p0,q0)-forms in a neighborhood of b for some fixed 0 ≤ p0 ≤ n, 1 ≤ q0 ≤ n, or M is a K\"ahler metric whose holomorphic bisectional curvature acting (p,q)-forms is positive; and 2) there exists a family of vector fields Tε that are transverse to the boundary b and generate one forms, which when applied to (p,q)-forms, 0 ≤ p ≤ n and q0 ≤ q ≤ n, satisfy a "weak form" of the compactness estimate. We also provide examples and applications of our main theorems.