We investigate regularity properties of the $overline{partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseu
doconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
We present a method for constructing global holomorphic peak functions from local holomorphic support functions for broad classes of unbounded domains. As an application, we establish a method for showing the positivity and completeness of invariant
metrics including the Bergman metric mainly for the unbounded domains.
The Kohn-Nireberg domains are unbounded domains in the complex Euclidean space of dimension 2 upon which many outstanding questions are yet to be explored. The primary aim of this article is to demonstrate that the Bergman and Caratheodory metrics of
any Kohn-Nirenberg domains are positive and complete.
The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forellis theorem on the complex analyticity of the functions that are: (1) $mathcal{C}^infty$ smooth at a poi
nt, and (2) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.
The first result is the semicontinuity of automorphism groups for the collection of complex two-dimensional bounded pseudoconvex domains with smooth boundary of finite DAngelo type. The method of proof is new so that it simplifies the previous proof
of earlier semicontinuity theorems on bounded strongly pseudoconvex daomains by Greene and Krantz in the early 1980s.
We describe the boundary behaviors of the squeezing functions for all bounded convex domains in $mathbb{C}^n$ and bounded domains with a $C^2$ strongly convex boundary point.
