Smooth equivalence of deformations of domains in complex euclidean spaces

TL;DR

This paper investigates when smooth families of domains in complex spaces are smoothly equivalent, showing that equivalence under discontinuous biholomorphisms does not imply smooth equivalence, and providing conditions for smooth equivalence.

Contribution

It establishes conditions for smooth equivalence of domain families and constructs examples where discontinuous equivalence does not imply smooth equivalence.

Findings

Smooth equivalence is not guaranteed by discontinuous biholomorphisms.

Constructed examples of non-smoothly equivalent but biholomorphically equivalent domain families.

Provided sufficient conditions for smooth equivalence of domain families.

Abstract

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 $n\geq2$, 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.