We prove that two smooth families of 2-connected domains in $cc$ are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct, for $m geq 3$, two smooth families of smoothly bounded $m$-connected domains in $cc$, and for $ngeq2$, two families of strictly pseudoconvex domains in $cc^n$, that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains.
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 pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
