Based on some ideas of Greene and Krantz, we study
the semicontinuity of automorphism groups of domains in one
and several complex variables. We show that semicontinuity fails
for domains in

A domain in

A notable theorem of Greene and Krantz [

Let

In what follows we shall refer to this result as the “semicontinuity theorem.”

It should be noted that, although this theorem was originally proved for strongly pseudoconvex domains in

The original proof of this result, which was rather complicated, used stability results for the Bergman kernel and metric established in [

It is geometrically natural to wonder whether there is a semicontinuity theorem when the boundary has smoothness of degree less than 2. On the one hand, experience in geometric analysis suggests that

The purpose of this paper is to show that the semicontinuity theorem fails for domains in

The main result of this section is the following.

Let

for each

See [

It should be understood that all the domains considered in this paper have finite connectivity. In particular, the complement of the domain only has finitely many components. And each component of the complement has Lipschitz boundary. We do

We shall use some ideas in [

Let

We will build our domains by modifying the unit ball

We define

Now define

In general we let, for

Now define, for

Finally we let

Now it is clear that

That completes the construction described in the theorem.

The one-variable result is the following.

Consider domains in

We see here that the situation is in marked contrast to that for several complex variables. Our proof of this result will rely on uniformization for planar domains, a result which has no analogue in several complex variables.

Fix the domain

It is a standard result of classical function theory that a finitely connected domain in the plane, with no component of the complement equal to a point, is conformally equivalent to the plane with finitely many nontrivial closed discs excised—see [

Thus we may apply the one-dimensional version of the semicontintuity theorem for

That completes the proof.

We note that another approach to construct the normalization map is by way of Green’s functions. This method is also quite explicit and constructive. Stability results for elliptic boundary value problems are well known. So this again leads to a proof of the semicontinuity theorem by transference to the normalized domain.

Key to the work of Greene and Krantz in [

Let

This result also holds in one complex dimension, and the proof in that context is actually much easier.

Our remark now is that this theorem is actually true in the Lipschitz topology. We use the argument of the last section. Namely, if

A lovely result of Maskit [

Let

An elegant corollary of Maskit’s result is that if

We would like to remark here that the ideas in this paper give a “poor man’s version” of this theorem. For let

A classical result in several complex variables is the following (see [

Let

In a similar spirit, Krantz [

Let

In this section we will reexamine Theorem

If

If

If

If

Thus, we see by inspection that Theorem

It is a matter of considerable interest to know the curvature properties of the Bergman metric on a planar domain. In particular, negativity of the curvature near the boundary is a useful analytic tool (see [

It is natural to want to consider the results presented here in either the

In several complex variables, one would also like to prove semicontinuity theorems for broad classes of domains. This will be the subject for future papers.

^{n}by its automorphism group