REINHARDT DOMAINS WITH

NON-COMPACT AUTOMORPHISM GROUPS

Siqi Fu Alexander V. Isaev Steven G. Krantz

Abstract. We give an explicit description of smoothly bounded Reinhardt domains with noncompact automorphism groups. In particular, this description confirms a special case of a conjecture of Greene/Krantz.

0. Introduction

Let D be a bounded domain in ℂ^{n}, n ≥ 2. Denote by Aut(D) the group of holomorphic automorphisms of D. The group Aut(D) with the topology of uniform convergence on compact subsets of D is in fact a Lie group (see [Ko]).

This paper is motivated by known results characterizing a domain by its automorphism group (see e.g. [R], [W], [BP]). More precisely, we assume that Aut(D) is not compact, i.e. there exist p ∈ D, q ∈ ∂D and a sequence {F_{i}} in Aut(D) such that F_{i}(p) → q as i → ∞. A point q ∈ ∂D with the above property is called a boundary accumulation point for Aut(D). An important issue for describing a domain D in terms of Aut(D) is the geometry of ∂D near a boundary accumulation point q (see e.g. [BP], [GK2]). In particular, we will be interested in the type of ∂D at q in the sense of D’Angelo [D’A], which measures the order of contact that complex varieties passing through q may have with ∂D.

We note in passing that it is known that ∂D must be pseudoconvex at a boundary accumulation point [GK1]. It is desirable to have additional geometric information about boundary accumulation points. We will be discussing the following conjecture that can be found in [GK2]: if D is a bounded domain in ℂ^{n} with C^{∞}-smooth boundary and if q is a boundary accumulation point for Aut(D), then q is a point of finite type.

For convex domains the conjecture was studied in [Ki]. Here we assume that D is a Reinhardt domain, i.e. a domain which the standard action of the n-dimensional torus 𝕋^{n} on ℂ^{n},

z_{j} ↦ e^{iϕ^{j}}z_{j},  ϕ_{j} ∈ ℝ, j = 1, …, n,  (0.1)

leaves invariant. The automorphism groups of bounded (and even hyperbolic) Reinhardt domains have been determined in [Su], [Sh2], [Kr]. We will use this description to prove the following classification result.

Typeset by AMS-TEX

Mathematics Subject Classification: 32A07, 32H05, 32M05

Key words and phrases. Automorphism groups, Reinhardt domains, domains of finite type..

Research at MSRI supported in part by NSF grant #DMS 9022140.

Theorem. If D is a bounded Reinhardt domain in ℂ^{n} with C^{∞}-smooth boundary, and if Aut(D) is not compact, then, up to dilations and permutations of coordinates, D is a domain of the form

{|z^{1}|^{2} + P (|z^{2}|, …, |z^{p}|) < 1} ,  (0.2)

where P is a polynomial:

P (|z^{2}|, …, |z^{p}|) = ∑_{j=2}^{p} r^{j}|z^{j}|^{2m_{j}} + ∑_{l_{2},...,l_{p}} a_{l_{2},...,l_{p}}|z^{2}|^{2l_{2}} … |z^{p}|^{2l_{p}},

r^{j} > 0, a_{l_{2},...,l_{p}} are real parameters, m_{j} ∈ ℕ, with the sum taken over all (p − 1)-tuples (l_{2}, …, l_{p}), l_{j} ∈ ℕ, such that ∑_{j=2}^{p} \frac{l_{j}}{m_{j}} = 1, and the complex variables z_{1}, …, z_{n} are divided into p non-empty groups z^{1}, …, z^{p}.

Remark. Domains (0.2) are a special case of an example in [BP]. The above theorem also corrects a conjecture of Catlin and Pinchuk (see [Kra]).

A byproduct of (the proof of) the results presented here is some information about the size of the set of boundary accumulation points. For example, one might ask whether boundary accumulation points can be isolated, or whether they can form a relatively open set in the boundary. Our calculations show that, for a bounded C^{1}-Reinhardt domain, the set S of boundary accumulation points form a manifold of dimension between 1 and 2n − 1 inclusive. The case of dim S = 2n − 1 (or top dimension) only occurs when the domain under consideration is the ball (up to dilations and permutations of the coordinates).

Note that the domains described in the theorem are not necessarily pseudoconvex. Indeed, consider the following example:

Example 1. Let D be the following bounded domain in ℂ^{3}

D = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} : |z_{1}|^{2} + |z_{2}|^{4} + |z_{3}|^{4} − |z_{2}|^{2}|z_{3}|^{2} < 1} .

To show that D has noncompact automorphism group consider the following sequence of automorphisms {F_{i}}

z_{1} ↦ \frac{z_{1} − a_{i}}{1 − \overline{a_{i}}z_{1}} ,

z_{2} ↦ \frac{(1 − |a_{i}|^{2})^{\frac{1}{4}} z_{2}}{\sqrt{1 − \overline{a_{i}}z_{1}}} ,

z_{3} ↦ \frac{(1 − |a_{i}|^{2})^{\frac{1}{4}} z_{3}}{\sqrt{1 − \overline{a_{i}}z_{1}}} ,

where |a_{i}| < 1, a_{i} → −1 as i → ∞, and the point p = (0, 0, 0) ∈ D. Then F_{i}(p) = (−a_{i}, 0, 0) → (1, 0, 0) ∈ ∂D. Consider next the boundary point q̃ = (\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt[4]{2}}). The complex tangent space at q̃ is

\{(z_{1}, z_{2}) : z_{1} + 2^{\frac{3}{4}} z_{3} = 0\},

and the Levi form at q̃ is

\frac{1}{\sqrt{2}}(−|z_{2}|^{2} + 8|z_{3}|^{2}),

which is clearly not non-negative. Note that q̃ is not a boundary accumulation point. □

Since domains (0.2) have real-analytic boundaries, [DF] implies the following result.

Corollary. If D is a smoothly bounded Reinhardt domain and Aut(D) is noncompact, then D is of finite type. In particular, the Greene/Krantz conjecture holds for D.

Despite the fact that the description of the automorphism groups of hyperbolic Reinhardt domains essentially coincides with that of bounded Reinhardt domains, the Greene/Krantz conjecture fails for the hyperbolic case:

Example 2. Define D ⊂ ℂ^{2} as follows

D = {(z_{1}, z_{2}) ∈ ℂ^{2} : |z_{1}|^{2} + (1 − |z_{1}|^{2})^{2}|z_{2}|^{2} < 1}.

The domain D is smooth and hyperbolic (see e.g. [PS]). Further, Aut(D) is noncompact. Indeed, consider the sequence of automorphisms {F_{i}}

\begin{matrix}z_{1} ↦ \frac{z_{1} − a_{i}}{1 − \overline{a_{i}}z_{1}} , \\ z_{2} ↦ \frac{(1 − \overline{a_{i}}z_{1})z_{2}}{\sqrt{1 − |a_{i}|^{2}}} , \end{matrix}

where, as above, |a_{i}| < 1, a_{i} → −1 as i → ∞, and the point p = (0, 0) ∈ D. Then we have F_{i}(p) = (−a_{i}, 0) → (1, 0) ∈ ∂D. Here q = (1, 0) is a boundary accumulation point of infinite type since it belongs to the complex affine subspace {z_{1} = 1} that entirely lies in ∂D.

Note that in this example ∂D is pseudoconvex at the boundary accumulation point q, but not globally pseudoconvex. □

Before proceeding, we wish to thank Nikolai Kruzhilin for a fruitful discussion. We note that this work was initiated while the third author was the (visiting) Richardson Fellow at the Australian National University. The third author was also supported by a grant from the National Science Foundation of the United States.

1. Proof of Theorem

Following [Sh2] we denote by Aut_{alg}((ℂ^{∗})^{n}) the group of algebraic automorphisms of (ℂ^{∗})^{n}, i.e. the group of mappings of the form

z_{i} ↦ λ_{i}z_{1}^{a_{i1}} … z_{n}^{a_{in}}, i = 1, … n,  (1.1)

where λ_{i} ∈ ℂ^{∗}, a_{ij} ∈ ℤ, and det(a_{ij}) = ±1. Here ℂ^{∗} = ℂ \ {0}.

For a bounded Reinhardt domain D ⊂ ℂ^{n}, denote by Aut_{alg}(D) the subgroup of Aut(D) that consists of algebraic automorphisms of D, i.e. automorphisms induced by mappings from Aut_{alg}((ℂ^{∗})^{n}). It is shown in [Sh2], [Kr] that Aut(D) = Aut_{0}(D) · Aut_{alg}(D), where Aut_{0}(D) is the identity component of Aut(D) and the dot denotes the composition operation in Aut(D). We will prove the main theorem using the explicit description of Aut_{0}(D) given in [Su], [Sh2], [Kr] together with the following proposition.

Proposition 1.1. For a smoothly bounded Reinhardt domain D ⊂ ℂ^{n}, Aut_{alg}(D) is compact.

Remark. The proposition will be proved in Section 2. In fact, we will show there that Aut_{alg}(D) is finite up to the action of 𝕋^{n} (see (0.1)). A plausible conjecture is that this last statement is true for any bounded Reinhardt domain in ℂ^{n}.

Corollary 1.2. For a smoothly bounded Reinhardt domain D ⊂ ℂ^{n}, Aut(D) is non-compact iff Aut_{0}(D) is non-compact.

We will now present the description of Aut_{0}(D) from [Su], [Sh2], [Kr]. Any bounded Reinhardt domain in ℂ^{n} can—by a biholomorphic mapping of the form (1.1)—be put into a normalized form G written as follows. There exist integers 0 ≤ s ≤ t ≤ p ≤ n and n_{i} ≥ 1, i = 1, …, p, with ∑_{i=1}^{p} n_{i} = n, and nonnegative real numbers α_{i}^{j}, i = 1, …, s, j = s + 1, …, p, with α_{i}^{j} = 0 for i = 1, …, s, j = s + 1, …, t, such that if we set z^{i} = (z_{n_{1}+···+n_{i−1}+1}, …, z_{n_{1}+···+n_{i}}), i = 1, …, p, then G̃ := G ⋂ {z^{i} = 0, i = 1, …, t} is a bounded Reinhardt domain in ℂ^{n_{t+1}} × ⋯ × ℂ^{n_{p}}, and G can be written in the form

\begin{matrix}G = \{|z^{1}| & < 1, …, |z^{s}| < 1, \\ (\frac{z^{t+1}}{∏_{i=1}^{s} (1 − |z^{i}|^{2})^{α_{i}^{t+1}} ∏_{j=s+1}^{t} exp (−β_{j}^{t+1} |z^{j}|^{2})} , …, \\ \frac{z^{p}}{∏_{i=1}^{s} (1 − |z^{i}|^{2})^{α_{i}^{p}} ∏_{j=s+1}^{t} exp (−β_{j}^{p} |z^{j}|^{2})}) ∈ G̃\}, \end{matrix}  (1.2)

for some non-negative β_{j}^{k}, j = s + 1, …, t, k = t + 1, …, p. A normalized form can be chosen so that Aut_{0}(G) is given by the following formulas:

\begin{matrix}z^{i} ↦ \frac{A^{i}z^{i} + b^{i}}{c^{i}z^{i} + d^{i}} , i = 1, …, s, \\ z^{j} ↦ B^{j}z^{j} + e^{j}, j = s + 1, …, t, \\ z^{k} ↦ C^{k} \frac{∏_{j=s+1}^{t} exp (−β_{j}^{k} (2\overline{e^{j}}^{T} B^{j}z^{j} + |e^{j}|^{2})) z^{k}}{∏_{i=1}^{s}(c^{i}z^{i} + d^{i})^{2α_{i}^{k}}} , k = t + 1, …, p, \end{matrix} (1.3)

where

( \begin{matrix}A^{i} & b^{i} \\ c^{i} & d^{i} \end{matrix} ) ∈ SU(n_{i}, 1), i = 1, …, s,

B^{j} ∈ U(n_{j}), e^{j} ∈ ℂ^{n_{j}}, j = s + 1, …, t,

C^{k} ∈ U(n_{k}), k = t + 1, …, p.

The above classification implies that Aut_{0}(G) is non-compact only if t > 0.

We are now going to select only those normalized forms (1.2) with t > 0 that can be the images of bounded domains with C^{∞}-boundaries under mappings of the form (1.1). We will need the following sequence of lemmas.

Lemma 1.3. Let D be a smoothly bounded Reinhardt domain and H_{k_{1},...,k_{r}} a coordinate subspace

H_{k_{1},...,k_{r}} = ⋂_{j=1}^{r}{z_{k_{j}} = 0}, r < n,

such that ∂D ∩ H_{k_{1},...,k_{r}} ≠ ∅. Then D ∩ H_{k_{1},...,k_{r}} is a nonempty smoothly bounded set in H_{k_{1},...,k_{r}}.

Proof. First we prove the lemma for one coordinate hyperplane H_{k} = {z_{k} = 0}. We will show that H_{k} may only intersect ∂D transversally. Indeed, assume that for some point q = (q_{1}, …, q_{k−1}, 0, q_{k+1}, …, q_{n}) ∈ ∂D, H_{k} coincides with the complex tangent space to ∂D at q. Consider the affine complex line S = {z_{j} = q_{j}|j ≠ k} that intersects ∂D at q transversally. Then D ∩ S is a smooth domain in S near q. On the other hand, there exists r > 0 such that, for every 0 < ρ < r, D ∩ S contains a point (q_{1}, …, q_{k−1}, z_{k}, q_{k+1}, …, q_{n}) with |z_{k}| = ρ. Since D is invariant under rotations in z_{k} it follows that, near q, D ∩ S coincides with the punctured disk {0 < |z_{k}| < r}, and therefore is not smooth.

Hence, H_{k} intersects ∂D transversally everywhere, and D ∩ H_{k} is a nonempty smoothly bounded set which is a finite collection of Reinhardt domains in H_{k}. An inductive argument now completes the proof. □

Lemma 1.4. If G is a normalized form of a smoothly bounded Reinhardt domain D, then s = t.

Proof. If t > s, then the normalized form G is unbounded in the z^{s+1}, …, z^{t}- directions (see (1.3)). Since D is bounded and G is obtained from D by a mapping of the form (1.1) it follows that, for some i_{0},

D ∩ \{z_{i_{0}} = 0\} = ∅,  and  \overline{D} ∩ \{z_{i_{0}} = 0\} ≠ ∅,

which is impossible by Lemma 1.3. □

Lemma 1.5. Let G be a normalized form of a smoothly bounded Reinhardt domain D.

Then if p > s, the following holds:

(i) For every s + 1 ≤ j ≤ p there exists 1 ≤ i ≤ s such that α_{i}^{j} > 0; (ii) s = 1; (iii) G contains the origin.

If p = s, then G is the unit ball.

Proof. Let p > s. Suppose first that α_{i}^{j} = 0 for all i = 1, …, s and j = s + 1, …, p. Then G is the direct product

G = {|z^{1}| < 1} × ⋯ × {|z^{s}| < 1} × G̃,

and therefore cannot be biholomorphically equivalent to a smoothly bounded domain [HO].

Renumbering the coordinates if necessary, we assume now that there exists s < k ≤ p such that, for every s + 1 ≤ j ≤ k, there is 1 ≤ i(j) ≤ s with α_{i(j)}^{j} > 0, and α_{i}^{j} = 0 for j = k + 1, …, p, i = 1, …, s. Choose a sequence of points (z_{l}^{1}, …, z_{l}^{s}, z_{l}^{s+1}, …, z_{l}^{p}) in G such that for all indices 1 ≤ i ≤ s, |z_{l}^{i}| → 1 as l → ∞. Since the domain G̃ is bounded, this implies that z_{l}^{j} → 0 for j = s + 1, …, k. Therefore, ∂G intersects the coordinate subspace H_{n_{s}+1,...,M(k)}, where M(k) = ∑_{j=s+1}^{k} n_{j}. Let q ∈ ∂G ∩ H_{n_{s}+1,...,M(k)} and let F = (F_{1}, …, F_{n}) denote the normalizing mapping for D. It now follows that there exists a sequence {q_{l}} in D such that F (q_{l}) → q, and therefore |F^{j}(q_{l})| → 1 for j = 1, …, s, F_{j}(q_{l}) → 0 for j = n_{s} + 1, …, M(k). Since D is smoothly bounded, Lemma 1.3 implies that there exists a coordinate subspace H_{k_{1},...,k_{r}}, r < n, such that D∩H_{k_{1},...,k_{r}} ≠ ∅, and F_{j} ≡ 0 for j = n_{s} + 1, …, M(k) on D ∩ H_{k_{1},...,k_{r}}. Therefore G_{k} = G ∩ H_{n_{s}+1,...,M(k)} ≠ ∅. At the same time G_{k} is the direct product

G_{k} = {|z^{1}| < 1} × ⋯ × {|z^{s}| < 1} × {G̃ ∩ H_{n_{s}+1,...,M(k)}}.  (1.4)

This domain is algebraically equivalent to F^{−1}(G_{k}), which is a smoothly bounded set by Lemma 1.3. The direct product in (1.4) is nontrivial if k < p or s > 1. Therefore, in these cases (as above, for k = s) we get a contradiction. Further, since G_{p} ≠ ∅, G contains the origin.

For p = s, G is the direct product

G = {|z^{1}| < 1} × ⋯ × {|z^{s}| < 1},

which by the same argument must be trivial, i.e. s = 1. Hence, G is the unit ball. The lemma is proved. □

By Lemma 1.5, G contains the origin; therefore the normalizing mapping for D is of the form

z_{i} ↦ λ_{i}z_{σ(i)},

where λ_{i} ∈ ℂ^{∗} and σ is a permutation of {1, …, n} [Su], [Sh1]. Therefore, to prove the theorem it is sufficient to consider domains of the form

G = \{|z^{1}| < 1, (\frac{z^{2}}{(1 − |z^{1}|^{2})^{α^{2}}} , …, \frac{z^{p}}{(1 − |z^{1}|^{2})^{α^{p}}} ) ∈ G̃\} ,  (1.5)

where G̃ is a bounded Reinhardt domain in ℂ^{n_{2}} × ⋯ × ℂ^{n_{p}} containing the origin, α^{j} > 0, j = 2, …, p.

Lemma 1.6. If a domain of the form (1.5) is smoothly bounded and if p ≥ 2, then for j = 2, …, p, α^{j} = \frac{1}{2m_{j}}, m_{j} ∈ ℕ, and G ⋂ (∩_{i=2,i≠j}^{p}\{z^{i} = 0\}) has the form

{|z^{1}|^{2} + r^{j}|z^{j}|^{2m_{j}} < 1} , r^{j} > 0.

Proof. First we observe that, by Lemma 1.3, G̃ is a smoothly bounded domain. Next, fix 2 ≤ j ≤ p. Analogously, since G̃ is smooth, then G̃ ⋂(∩_{i=2,i≠j}^{p}{z^{i} = 0}) is smooth. Further, G̃ ⋂ (∩_{i=2,i≠j}^{p}{z^{i} = 0}) is invariant under unitary transformations in z^{j} (see (1.3)). This implies that

G̃ ⋂ (∩_{i=2,i≠j}^{p}{z^{i} = 0}) = {|z^{j}| < A} ⋃ (∪_{l=1}^{k}{ρ_{l} < |z^{j}| < ν_{l}})  (1.6)

for some A, k, ρ_{1}, …, ρ_{k}, ν_{1}, …, ν_{k}. Therefore

\begin{aligned}G ⋂ (∩_{i=2,i≠j}^{p}\{z^{i} = 0\}) & = \{|z^{1}|^{2} + \frac{1}{A^{\frac{1}{α^{j}}}} |z^{j}|^{\frac{1}{α^{j}}} < 1\} ⋃ \\ & ⋃_{l=1}^{k}\{ρ_{l}(1 − |z_{1}|^{2})^{α^{j}} < |z^{j}| < ν_{l}(1 − |z_{1}|^{2})^{α^{j}}\}. \end{aligned}

(1.7)

Since G is smooth, G ⋂ (∩_{i=2,i≠j}^{p}{z^{i} = 0}) is smooth. Together with (1.7) this shows that α^{j} = \frac{1}{2m_{j}}, m_{j} ∈ ℕ, and that in (1.6) one in fact has

G̃ ⋂ (∩_{i=2,i≠j}^{p}{z^{i} = 0}) = {|z^{j}| < A}.

This proves the lemma. □

The main step in the proof of the theorem is the following proposition. Proposition 1.7. Let G be a smoothly bounded domain of the form (1.5). Then G is given by

G = {|z^{1}|^{2} + P (|z^{2}|, …, |z^{p}|) < 1} .

Here P is a polynomial of the form

P (|z^{2}|, …, |z^{p}|) = ∑_{j=2}^{p} r^{j}|z^{j}|^{2m_{j}} + ∑_{l_{2},...,l_{p}} a_{l_{2},...,l_{p}}|z^{2}|^{2l_{2}} … |z^{p}|^{2l_{p}},  (1.8)

where r^{j} > 0, a_{l_{2},...,l_{p}} ∈ ℝ and the sum is taken over all (p − 1)-tuples (l_{2}, …, l_{p}), l_{j} ∈ ℕ, such that ∑_{j=2}^{p} \frac{l_{j}}{m_{j}} = 1. Proof. If p = 1, then by Lemma 1.5 we see that G is the unit ball. Assume now that p ≥ 2. We write G, near q = (1, 0, …, 0) ∈ ∂G, in the form

|z_{1}|^{2} + ϕ(z_{2}, …, z_{n}) < 1,  (1.9)

where ϕ is a smooth function in a neighbourhood of the origin in ℂ^{n−1}, ϕ(0) = 0, grad ϕ(0) = 0. Since G is invariant under unitary transformations in each of z^{1}, … z^{p}, (see (1.3)), equation (1.9) is equivalent to

|z^{1}|^{2} + ψ(|z^{2}|, …, |z^{p}|) < 1,  (1.10)

where

ψ(|z^{2}|, …, |z^{p}|) = ϕ( \overbrace{0, …, 0}^{n_{1}−1 times}; |z^{2}|, \overbrace{0, …, 0}^{n_{2}−1 times}; …; |z^{p}|, \overbrace{0, …, 0}^{n_{p}−1 times}).

Consider the following family of automorphisms of G:

z_{1} ↦ \frac{z_{1} − a}{1 − az_{1}} ,

z_{i} ↦ \frac{\sqrt{1 − a^{2}}z_{i}}{1 − az_{1}} , i = 2, …, n_{1},

z^{j} ↦ \frac{(\sqrt{1 − a^{2}})^{\frac{1}{m_{j}}} z^{j}}{(1 − az_{1})^{\frac{1}{m_{j}}}} , j = 2, …, p,

where a is a non-negative parameter close to zero. These automorphisms are holomorphic in a neighbourhood of \overline{G} and map ∂G near q into itself. Therefore (1.10) gives that, on ∂G,

\frac{|z_{1} − a|^{2}}{|1 − az_{1}|^{2}} + ∑_{i=2}^{n_{1}} \frac{(1 − a^{2})|z_{i}|^{2}}{|1 − az_{1}|^{2}} + ψ (\frac{(\sqrt{1 − a^{2}})^{\frac{1}{m_{2}}} |z^{2}|}{|1 − az_{1}|^{\frac{1}{m_{2}}}}, …, \frac{(\sqrt{1 − a^{2}})^{\frac{1}{mp}} |z^{p}|}{|1 − az_{1}|^{\frac{1}{mp}}}) = 1.

It then follows that, on ∂G,

ψ (\frac{(\sqrt{1 − a^{2}})^{\frac{1}{m_{2}}} |z^{2}|}{|1 − az_{1}|^{\frac{1}{m_{2}}}}, …, \frac{(\sqrt{1 − a^{2}})^{\frac{1}{mp}} |z^{2}|}{|1 − az_{1}|^{\frac{1}{mp}}}) = \frac{1 − a^{2}}{|1 − az_{1}|^{2}} ψ(|z^{p}|, …, |z^{p}|).

This implies that

ψ(t^{\frac{1}{2m_{2}}} |z^{2}|, …, t^{\frac{1}{2mp}} |z^{p}|) = tψ(|z^{2}|, …, |z^{p}|),  (1.11)

for (z^{2}, …, z^{p}) in a neighbourhood of the origin and 1 ≤ t ≤ 1 + ϵ for some small ϵ > 0.

We will now prove that the homogeneity property (1.11) implies that ψ(|z^{2}|, …, |z^{p}|) has the form (1.8).

Lemma 1.8. Let f(x_{1}, …, x_{r}) be a C^{∞}-function in a neighbourhood of the origin in ℝ^{r}. Suppose that there exist k_{j} ∈ ℕ, j = 1, …, r, such that

f(t^{\frac{1}{k_{1}}} x_{1}, …, t^{\frac{1}{kr}} x_{r}) = tf(x_{1}, …, x_{r}),  (1.12)

for 1 ≤ t ≤ 1 + ϵ. Then f has the form

f(x_{1}, …, x_{r}) = ∑_{l_{1},...,l_{r}} b_{l_{1},...,l_{r}}x_{1}^{l_{1}} … x_{s}^{l_{r}},  (1.13)

where b_{l_{1},...,l_{r}} ∈ ℝ, and the sum is taken over all r-tuples (l_{1}, …, l_{r}), l_{j} ∈ ℤ, l_{j} ≥ 0, such that ∑_{j=1}^{r} \frac{l_{j}}{k_{j}} = 1.

Proof. Differentiating (1.12) at the origin with respect to x_{1}, …, x_{r}, we get

[t^{(∑{r \atop j=1} \frac{q_{j}}{k_{j}} )}]\frac{∂^{q_{1}+···+q_{r}}f}{∂x_{1}^{q_{1}} … ∂x_{r}^{q_{r}}} (0) = t\frac{∂^{q_{1}+···+q_{r}}f}{∂x_{1}^{q_{1}} … ∂x_{r}^{q_{r}}} (0),

which implies that \frac{∂^{q_{1}+···+qr} f}{∂x_{1}^{q_{1}} ...∂x_{r}^{qr}} (0) may be nonzero only if ∑_{j=1}^{r} \frac{q_{j}}{k_{j}} = 1. Therefore the Taylor formula for f in a neighbourhood of the origin gives that f = P + α, where P is a polynomial as in (1.13), and α is a C^{∞}-function in a neighbourhood of the origin satisfying (1.12) and such that

α(x) = o (|x|^{N})  (1.14)

for all sufficiently large N, as x → 0.

To show that α ≡ 0, we restrict α to the curve

x_{j}(u) = c_{j}u^{k_{1}...k_{j−1}k_{j+1}...k_{r}}, j = 1, …, r,  (1.15)

where u is a real parameter close to zero, c = (c_{1}, …, c_{r}) ∈ ℝ^{r}, |c| = 1. We denote this restriction by g_{c}(u). Then (1.12) gives

g_{c}(t^{\frac{1}{k_{1}...kr}} u) = tg_{c}(u).

Differentiating the last equality with respect to t and setting t = 1 we get

\frac{u}{k_{1} … k_{r}} g′_{c}(u) = g_{c}(u).

Solving this equation we obtain

g_{c}(u) = A(c)u^{k_{1}...k_{r}},

where A(c) ∈ ℝ. Further, (1.14) immediately implies that A(c) = 0, and since curves of the form (1.15) for all c cover a neighbourhood of the origin, it follows that α(x) ≡ 0.

The lemma is proved. □

Property (1.11), Lemma 1.8 and Lemma 1.6 immediately give that ψ(|z^{2}|, …, |z^{p}|) has the form (1.8). We will now show that equation (1.10) in fact defines G globally, not just in a neighbourhood of q. Indeed, fix 1 − |z^{1}|^{2} = δ, where δ is small. Then, (1.5) and (1.10) imply that G̃ is given by

G̃ = \{(z^{2}, …, z^{p}): ψ(δ^{\frac{1}{2m_{2}}} |z^{2}|, …, δ^{\frac{1}{2mp}} |z^{p}|) < δ\} .

It now follows from the homogeneity property (1.11) that

G̃ = {(z^{2}, …, z^{p}): ψ(|z^{2}|, …, |z^{p}|) < 1} .

This completes the proof of the proposition and the theorem. □

2. Proof of Proposition 1.1

Assume that

\begin{matrix}D ∩ \{z_{i} = 0\} ≠ ∅, i = 1, …, k, \\ D ∩ \{z_{i} = 0\} = ∅, i = k + 1, …, n, \end{matrix}

where 0 ≤ k ≤ n. For k = n it is shown in [Sh2] that Aut_{alg}(D) is finite up to the action of 𝕋^{n}, and is therefore compact.

Let k < n. Since D is smooth, by Lemma 1.3 we have

dist(D, {z_{i} = 0}) > 0, i = k + 1, … n.  (2.1)

By [Sh2], every algebraic automorphism of D has the form

\begin{matrix}z_{i} ↦ λ_{i}z_{σ(i)}z_{k+1}^{a_{i k+1}} … z_{n}^{a_{i n}}, i = 1, …, k, \\ z_{i} ↦ λ_{i}z_{k+1}^{b_{i k+1}} … z_{n}^{b_{i n}}, i = k + 1, …, n, \end{matrix}  (2.2)

where λ_{i} ∈ ℂ^{∗}, a_{ij} ∈ ℤ, b_{ij} ∈ ℤ, det(b_{ij}) = ±1, and σ is a permutation of {1, …, k}.

Consider the logarithmic image of D, i.e. the domain in ℝ^{n}(x_{1}, …, x_{n}) defined as

D_{log} = {(log |z_{1}|, …, log |z_{n}|) ∈ ℝ^{n} : (z_{1}, …, z_{n}) ∈ D, z_{1} … z_{n} ≠ 0} .

Since D is bounded, we can assume that it lies in the polydisk {|z_{1}| < 1, …, |z_{n}| < 1}. Then D_{log} ⊂ ℝ_{−}^{n}, where ℝ_{−}^{n} = {x_{1} < 0, …, x_{n} < 0}. The mappings (2.2) on D_{log} now become affine mappings of the form

( x_{1} ; ⋮ ; x_{k} ; x_{k+1} ; ⋮ ; x_{n} ) ↦ ( x_{σ(1)} ; ⋮ ; x_{σ(k)} ; 0 ; ⋮ ; 0 ) + ( \begin{matrix}\overbrace{0 … 0}^{k times} & a_{1 k+1} & … & a_{1 n} \\ ⋮ … ⋮ & ⋮ & … & ⋮ \\ 0 … 0 & a_{k k+1} & … & a_{k n} \\ 0 … 0 & b_{k+1 k+1} & … & b_{k+1 n} \\ ⋮ … ⋮ & ⋮ & … & ⋮ \\ 0 … 0 & b_{n k+1} & … & b_{n n} \end{matrix} ) ( 0 ; ⋮ ; 0 ; x_{k+1} ; ⋮ ; x_{n} ) + ( μ_{1} ; ⋮ ; μ_{k} ; μ_{k+1} ; ⋮ ; μ_{n} ) ,

(2.3)

where μ_{j} = log |λ_{j}|, j = 1, …, n. Let D′_{log} denote the projection of D_{log} to the subspace of the last n − k coordinates ℝ^{n−k} = {x_{1} = ⋯ = x_{k} = 0}. Property (2.1) implies that D′_{log} is a bounded subset of ℝ_{−}^{n−k}. Further, for any affine automorphism of D_{log} of the form (2.3), the mapping

x′ ↦ (b_{ij}) x′ + μ′,  (2.4)

where x′ = (x_{k+1}, …, x_{n}), μ′ = (μ_{k+1}, …, μ_{n}), is an automorphism of D′_{log}. Since D′_{log} is bounded, the group Aff(D′_{log}) of all affine automorphisms of D′_{log} is clearly compact. Since the b_{ij} are integers, the group of affine transformations of D′_{log} of the form (2.4) is closed in Aff(D′_{log}), and therefore is compact. This, in fact, implies that there are only finitely many transformations of D′_{log} of the form (2.4). Indeed, let

ψ_{m}(x′) = B_{m}x′ + μ′_{m}

be a sequence of distinct transformations of the form (2.4). Then, by choosing a convergent subsequence {ψ_{m_{l}}} and taking into account that the B_{m_{l}} are integer matrices, we conclude that B_{m_{l_{1}}} = B_{m_{l_{2}}} for large m_{l_{1}}, m_{l_{2}}. This implies that μ′_{m_{l_{1}}} = μ′_{m_{l_{2}}} since otherwise D′_{log} would be invariant under the translation

x′ ↦ x′ + μ′_{m_{l_{1}}} − μ′_{m_{l_{2}}} ,

which is impossible because D′_{log} is bounded. Therefore, the ψ_{m_{l}} become equal to each other for large m_{l}, which contradicts our choice of {ψ_{m}}.

Now we assume that k ≥ 1 and will show that if for two affine automorphisms F_{1}, F_{2} of D_{log} of the form (2.3) the induced automorphisms (2.4) of D′_{log} coincide, then F_{1} coincides with F_{2} up to a permutation of the first k components. Indeed, consider the automorphism F = F_{1}^{−1} ∘ F_{2}. For F the corresponding automorphism of D′_{log} is the identity. Therefore, if the matrix (a_{ij}) or the vector (μ_{1}, …, μ_{k}) is nonzero, then by iterating either F or F^{−1}, one can take a point from D_{log} outside ℝ_{−}^{n} by making one of its first k coordinates positive. Thus, for the automorphism F, a_{ij} = 0 and μ_{i} = 0 for all i, j, i.e. F_{1} differs from F_{2} by a permutation of the first k components.

Hence Aut_{alg}(D) is always finite up to the action of 𝕋^{n} and therefore is compact. The proposition is proved. □

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Siqi Fu, Department of Mathematics, University of California, Irvine, CA 92717 USA E-mail address: sfu@math.uci.edu

A. V. Isaev, Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT 0200, Australia

E-mail address: Alexander.Isaev@anu.edu.au

S. G. Krantz, Department of Mathematics, Washington University, St. Louis, MO 63130, USA

E-mail address: sk@math.wustl.edu