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

TL;DR

This paper provides broad conditions under which the $ar ext{d}$-Neumann problem exhibits exact regularity on pseudoconvex manifolds, extending understanding of boundary behavior and regularity in complex analysis.

Contribution

It introduces new sufficient conditions involving plurisubharmonic functions and vector fields for establishing regularity in the $ar ext{d}$-Neumann problem on complex manifolds.

Findings

Established conditions for exact regularity on pseudoconvex domains.

Connected regularity to plurisubharmonic functions and curvature positivity.

Provided examples and applications illustrating the theoretical results.

Abstract

We establish general sufficient conditions for exact (and global) regularity in the $\bar\partial$-Neumann problem on $(p,q)$-forms, $0 \leq p \leq n$ and $1\leq q \leq n$, on a pseudoconvex domain $\Omega$ with smooth boundary $b\Omega$ 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 $b\Omega$ for some fixed $0 \leq p_0 \leq n$, $1 \leq q_0 \leq n$, 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 $T_\epsilon$ that are transverse to the boundary $b\Omega$ and generate one forms, which when applied to $(p,q)$-forms, $0 \leq p \leq n$ and $q_0 \leq q \leq n$, satisfy a "weak form" of the compactness estimate. We also provide examples and applications of our main theorems.