On proper R-actions on hyperbolic Stein surfaces

— In this paper we investigate proper R–actions on hyperbolic Stein surfaces and prove in particular the following result: Let D ⊂ C be a simply-connected bounded domain of holomorphy which admits a proper R–action by holomorphic transformations. The quotient D/Z with respect to the induced proper Z–action is a Stein manifold. A normal form for the domain D is deduced.


Introduction
Let X be a Stein manifold endowed with a real Lie transformation group G of holomorphic automorphisms.In this situation it is natural to ask whether there exists a G-invariant holomorphic map π : X → X//G onto a complex space X//G such that O X/ /G = (π * O X ) G and, if yes, whether this quotient X//G is again Stein.If the group G is compact, both questions have a positive answer as is shown in [Hei91].
For non-compact G even the existence of a complex quotient in the above sense of X by G cannot be guaranteed.In this paper we concentrate on the most basic and already non-trivial case G = R.We suppose that G acts properly on X.Let Γ = Z.Then X/Γ is a complex manifold and if, moreover, it is Stein, we can define X//G := (X/Γ)//(G/Γ).The following was conjectured by Alan Huckleberry.
Let X be a contractible bounded domain of holomorphy in C n with a proper action of G = R. Then the complex manifold X/Γ is Stein.
In [FI01] this conjecture is proven for the unit ball and in [Mie08] for arbitrary bounded homogeneous domains in C n .In this paper we make a first step towards a proof in the general case by showing Theorem.-Let D be a simply-connected bounded domain of holomorphy in C 2 .Suppose that the group R acts properly by holomorphic transformations on D. Then the complex manifold D/Z is Stein.Moreover, D/Z is biholomorphically equivalent to a domain of holomorphy in C 2 .
As an application of this theorem we deduce a normal form for domains of holomorphy whose identity component of the automorphism group is non-compact as well as for proper R-actions on them.Notice that we make no assumption on smoothness of their boundaries.
We first discuss the following more general situation.Let X be a hyperbolic Stein manifold with a proper R-action.Then there is an induced local holomorphic C-action on X which can be globalized in the sense of [HI97].The following result is central for the proof of the above theorem.
Theorem.-Let X be a hyperbolic Stein surface with a proper R-action.Suppose that either X is taut or that it admits the Bergman metric and H 1 (X, R) = 0. Then the universal globalization X * of the induced local C-action is Hausdorff and C acts properly on X * .Furthermore, for simply-connected X one has that X * → X * /C is a holomorphically trivial C-principal bundle over a simply-connected Riemann surface.
Finally, we discuss several examples of hyperbolic Stein manifolds X with proper R-actions such that X/Z is not Stein.If one does not require the existence of an Raction, there are bounded Reinhardt domains in C 2 with proper Z-actions for which the quotients are not Stein.

Hyperbolic Stein R-manifolds
In this section we present the general set-up.
2.1.The induced local C-action and its globalization.-Let X be a hyperbolic Stein manifold.It is known that the group Aut(X) of holomorphic automorphisms of X is a real Lie group with respect to the compact-open topology which acts properly on X (see [Kob98]).Let {ϕ t } t∈R be a closed one parameter subgroup of Aut(D).Consequently, the action R × X → X, t • x := ϕ t (x), is proper.By restriction, we obtain also a proper Z-action on X.Since every such action must be free, the quotient X/Z is a complex manifold.This complex manifold X/Z carries an action of S 1 ∼ = R/Z which is induced by the R-action on X.
Integrating the holomorphic vector field on X which corresponds to this R-action we obtain a local C-action on X in the following sense.There are an open neighborhood Ω ⊂ C × X of {0} × X and a holomorphic map Φ : Ω → X, Φ(t, x) =: t • x, such that the following holds: (1) For every x ∈ X the set Ω(x (2) for all x ∈ X we have 0 whenever both sides are defined.Following [Pal57] (compare [HI97] for the holomorphic setting) we say that a globalization of the local C-action on X is an open R-equivariant holomorphic embedding ι : X ֒→ X * into a (not necessarily Hausdorff) complex manifold X * endowed with a holomorphic C-action such that C Since X is Stein, the universal globalization X * of the induced local C-action exists as is proven in [HI97].We will always identify X with its image ι(X) ⊂ X * .Then the local C-action on X coincides with the restriction of the global C-action on X * to X.
Recall that X is said to be orbit-connected in X * if for every x ∈ X * the set Σ(x) := {t ∈ C; t • x ∈ X} is connected.The following criterion for a globalization to be universal is proven in [CTIT00].
Lemma 2.1.-Let X * be any globalization of the induced local C-action on X.Then X * is universal if and only if X is orbit-connected in X * .
Remark.-The results about (universal) globalizations hold for a bigger class of groups ([CTIT00]).However, we will need it only for the groups C and C * and thus will not give the most general formulation.
For later use we also note the following Lemma 2.2.-The C-action on X * is free.
Proof.-Suppose that there exists a point x ∈ X * such that C x is non-trivial.Because of C•X = X * we can assume that x ∈ X holds.Since C x is a non-trivial closed subgroup of C, it is either a lattice of rank 1 or 2, or C. The last possibility means that x is a fixed point under C which is not possible since R acts freely on X.
We observe that the lattice C x is contained in the connected R-invariant set Σ(x) = {t ∈ C; t • x ∈ X}.By R-invariance Σ(x) is a strip.Since X is hyperbolic, this strip cannot coincide with C. The only lattice in C which can possibly be contained in such a strip is of the form Zr for some r ∈ R. Since this contradicts the fact that R acts freely on X, the lemma is proven.
Note that we do not know whether X * is Hausdorff.In order to guarantee the Hausdorff property of X * , we make further assumptions on X.The following result is proven in [Ian03] and [IST04].
Theorem 2.3.-Let X be a hyperbolic Stein manifold with a proper R-action.Suppose in addition that X is taut or admits the Bergman metric.Then X * is Hausdorff.If X is simply-connected, then the same is true for X * .
We refer the reader to Chapter 4.10 and Chapter 5 in [Kob98] for the definitions and examples of tautness and the Bergman metric.
Remark.-Every bounded domain in C n admits the Bergman metric.
2.2.The quotient X/Z.-We assume from now on that X fulfills the hypothesis of Theorem 2.3.Since X * is covered by the translates t • X for t ∈ C and since the action of Z on each domain t • X is proper, we conclude that the quotient X * /Z fulfills all axioms of a complex manifold except for possibly not being Hausdorff.
We have the following commutative diagram: Note that the group Lemma 2.4.-The induced map X/Z ֒→ X * /Z is the universal globalization of the local C * -action on X/Z.
Proof.-The open embedding X ֒→ X * induces an open embedding X/Z ֒→ X * /Z.This embedding is S 1 -equivariant and we have C * • X/Z = X * /Z.This implies that X * /Z is a globalization of the local C * -action on X/Z.In order to prove that this globalization is universal, by the globalization theorem in [CTIT00] it is enough to show that X/Z is orbit-connected in X * /Z.Hence, we must show that for every For this we consider the set Σ(x) = {t ∈ C; t • x ∈ X}.Since the map X → X/Z intertwines the local C-and C * -actions, we conclude that t ∈ Σ(x) holds if and only if e 2πit ∈ Σ [x] holds.Since X * is universal, Σ(x) is connected which implies that Σ [x] is likewise connected.Thus X * /Z is universal.
Remark.-The globalization X * /Z is Hausdorff if and only if Z or, equivalently, R act properly on X * .As we shall see in Lemma 3.3, this is the case if X is taut.

2.3.
A sufficient condition for X/Z to be Stein.-If dim X = 2, we have the following sufficient condition for X/Z to be a Stein surface.
Proposition 2.5.-If the C-action on X * is proper and if the Riemann surface X * /C is not compact, then X/Z is Stein.
Proof.-Under the above hypothesis we have the C-principal bundle X * → X * /C.If the base X * /C is not compact, then this bundle is holomorphically trivial, i. e. X * is biholomorphic to C × R where R is a non-compact Riemann surface.Since R is Stein, the same is true for X * and for X * /Z ∼ = C * × R. Since X/Z is locally Stein, see [Mie08], in the Stein manifold X * /Z, the claim follows from [DG60].
Therefore, the crucial step in the proof of our main result consists in showing that C acts properly on X * under the assumption dim X = 2.

Local properness
Let X be a hyperbolic Stein R-manifold.Suppose that X is taut or that it admits the Bergman metric and H 1 (X, R) = {0}.We show that then C acts locally properly on X * .(1) For every x ∈ M the isotropy group G x is compact.
(3) The orbit space M/G is a smooth manifold which is in general not Hausdorff.
(4) All G-orbits are closed in M .
(5) The G-action on M is proper if and only if M/G is Hausdorff.
Proof.-The first claim is elementary to check.The second claim is proven in [DK00].
The third one is a consequence of (2) since the slices yield charts on M/G which are smoothly compatible because the transitions are given by the smooth action of G on M .Assertion (4) follows from (3) because in locally Euclidian topological spaces points are closed.The last claim is proven in [Pal61]. Remark.
-Since R acts properly on X, the R-action on X * is locally proper.

3.2.
Local properness of the C-action on X * .-Recall that we assume that (3.1) X is taut or that (3.2) X admits the Bergman metric and H 1 (X, R) = {0}.
We first show that assumption (3.1) implies that C acts locally properly on X * .Since X * is the universal globalization of the induced local C-action on X, we know that X is orbit-connected in X * .This means that for every x ∈ X * the set Σ(x) = {t ∈ C; t • x ∈ X} is a strip in C. In the following we will exploit the properties of the thickness of this strip.
Proof.-It is proven in [For96] that u and −o are plurisubharmonic on X.By equivariance, we obtain this result for X * .Now we prove that the function u : X \ {u = −∞} → R is continuous which was remarked without complete proof in [Ian03].For this let (x n ) be a sequence in X which converges to x 0 ∈ X \ {u = −∞}.Since u is upper semi-continuous, we have lim sup n→∞ u(x n ) ≤ u(x 0 ).Suppose that u is not continuous in x 0 .Then, after replacing (x n ) by a subsequence, we find ε > 0 such that u(x n ) ≤ u(x 0 ) − ε < u(x 0 ) holds for all n ∈ N. Consequently, we have Σ( Let us consider the sets The sets N (u) and P(u) are similarly defined.Since X = x ∈ X * ; u(x) < 0 < o(x) , we can recover X from X * with the help of u and o.
Lemma 3.3.-The action of R on X * is proper.
Proof.-Let ∂ * X denote the boundary of X in X * .Since the functions u and −o are continuous on X * \ P(u) and X * \ P(o) one verifies directly that ∂ * X = N (u) ∪ N (o) holds.As a consequence, we note that if x ∈ ∂ * X, then for every ε > 0 the element Let (t n ) and (x n ) be sequences in R and X * such that (t n • x n , x n ) converges to (y 0 , x 0 ) in X * × X * .We may assume without loss of generality that x 0 and hence x n are contained in X for all n.Consequently, we have y 0 ∈ X ∪ ∂ * X.If y 0 ∈ ∂ * X holds, we may choose an ε > 0 such that (i ε) • y 0 and (i ε) • x 0 lie in X.Since the R-action on X is proper, we find a convergent subsequence of (t n ) which was to be shown.Lemma 3.4.-We have: (1) N (u) and N (o) are R-invariant.
(2) We have (3) The sets P(u) and P(o) are closed, C-invariant and pluripolar in X * .
Proof.-The first claim follows from the R-invariance of u and o.
The second claim follows from u(x) < o(x).
The third one is a consequence of the R-invariance and iR-equivariance of u and o.
If there was a point x ∈ P(u) ∩ P(o), then C • x would be a subset of X which is impossible since X is hyperbolic.
is an iR-equivariant homeomorphism.Since R acts properly on N (o), it follows that C acts properly on X * \ P(o).The same holds when o is replaced by u.
Corollary 3.6.-The C-action on X * is locally proper.If P(o) = ∅ or P(u) = ∅ hold, then C acts properly on X * .
From now on we suppose that X fulfills the assumption (3.2).Recall that the Bergman form ω is a Kähler form on X invariant under the action of Aut(X).Let ξ denote the complete holomorphic vector field on X which corresponds to the Raction, i. e. we have ξ(x) = ∂ ∂t 0 ϕ t (x).Hence, Remark.-This means that µ ξ is a momentum map for the R-action on X.
implies that every iR-orbit intersects (µ ξ ) −1 (c) transversally.Since X is orbit-connected in X * , the map iR × (µ ξ ) −1 (c) → X * is injective and therefore a diffeomorphism onto its open image.Together with the fact that (µ ξ ) −1 (c) is R-invariant this yields the existence of differentiable local slices for the C-action.

3.3.
A necessary condition for X/Z to be Stein.-We have the following necessary condition for X/Z to be a Stein manifold.
Proposition 3.9.-If the quotient manifold X/Z is Stein, then X * is Stein and the C-action on X * is proper.
Proof.-Suppose that X/Z is a Stein manifold.By [CTIT00] this implies that X * is Stein as well.
Next we will show that the C * -action on X * /Z is proper.For this we will use as above a moment map for the S 1 -action on X * /Z.
By compactness of S 1 we may apply the complexification theorem from [Hei91] which shows that X * /Z is also a Stein manifold and in particular Hausdorff.Hence, there exists a smooth strictly plurisubharmonic exhaustion function ρ : where ξ is the complete holomorphic vector field on X * /Z which corresponds to the S 1 -action.Now we can apply the same argument as above in order to deduce that C * acts locally properly on X * /Z.We still must show that (X * /Z)/C * is Hausdorff.To see this, let C * • x j , j = 0, 1, be two different orbits in X * /Z.Since C * acts locally properly, these are closed and therefore there exists a function f ∈ O(X * /Z) with f | C * •x j = j for j = 0, 1. Again we may assume that f is S 1 -and consequently C * -invariant.Hence, there is a continuous function on (X * /Z)/C * which separates the two orbits, which implies that (X * /Z)/C * is Hausdorff.This proves that C * acts properly on X * /Z.
Since we know already that the C-action on X * is locally proper, it is enough to show that X * /C is Hausdorff.But this follows from the properness of the C * -action on X * /Z since X * /C ∼ = (X * /Z)/C * is Hausdorff.

Properness of the C-action
Let X be a hyperbolic Stein R-manifold.Suppose that X fulfills (3.1) or (3.2).We have seen that C acts locally properly on X * .In this section we prove that under the additional assumption dim X = 2 the orbit space X * /C is Hausdorff.This implies that C acts properly on X * if dim X = 2. 4.1.Stein surfaces with C-actions.-For every function f ∈ O(∆) which vanishes only at the origin, we define Since the differential of the defining equation of X f is given by f ′ (x)y f (x) − 2z , we see that 1 is a regular value of (x, y, z) There is a holomorphic C-action on X f defined by t • (x, y, z) := x, y + 2tz + t 2 f (x), z + tf (x) .
One can directly check that this defines an action.
Lemma 4.1.-The C-action on X f is free, and all orbits are closed.
Remark.-The orbit space X f /C is the unit disc with a doubled origin and in particular not Hausdorff.
We calculate slices at the point p j , j = 1, 2, as follows.Let ϕ j : ∆×C → X f be given by ϕ 1 (z, t) := t•(z, 0, i) and ϕ 2 (w, s) = s•(w, 0, −i).Solving the equation s•(w, 0, −i) = t • (z, 0, i) for (w, s) yields the transition function The function 1 f is a meromorphic function on ∆ without zeros and with the unique pole 0. Let U 1 ⊂ D be an R-invariant open neighborhood of p 1 .Then there are r, r We have to show that for all r 2 , r 3 > 0 there exist ( Let r 2 , r 3 > 0 be given.From (4.1) we obtain ε 3 = taf (ε 1 ) + 2i or, equivalently, ta = e for t = 0. Choosing a real number t ≫ 1, we find an ε 1 ∈ ∆ * r such that (4.2) is fulfilled.After possibly enlarging t we have ε 3 := taf (ε 1 ) + 2i = −2i ζ ta ∈ ∆ r 3 .Together with ε 2 = ε 2 equation (4.1) is fulfilled and the proof is finished.Thus, the Stein surface X f cannot be obtained as globalization of the local C-action on any R-invariant domain D ⊂ X f on which R acts properly.
4.2.The quotient X * /C is Hausdorff.-Suppose that X * /C is not Hausdorff and let x 1 , x 2 ∈ X be such that the corresponding C-orbits cannot be separated in X * /C.Since we already know that C acts locally proper on X * we find local holomorphic slices ϕ j : ∆×C → U j ⊂ X, ϕ j (z, t) = t•s j (z) at each C•x j where s j : ∆ → X is holomorphic with s j (0) = x j .Consequently, we obtain the transition function ϕ 12 : (∆ \ A) × C → (∆\A)×C for some closed subset A ⊂ ∆ which must be of the form (z, t) → z, t+f (z) for some f ∈ O(∆ \ A).The following lemma applies to show that A is discrete and that f is meromorphic on ∆.Hence, we are in one of the model cases discussed in the previous subsection.
Lemma 4.3.-Let ∆ 1 and ∆ 2 denote two copies of the unit disk {z ∈ C; |z| < 1}.Let U ⊂ ∆ j , j = 1, 2, be a connected open subset and f : U ⊂ ∆ 1 → C a non-constant holomorphic function on U .Define the complex manifold Suppose that M is Hausdorff.Then the complement A of U is discrete and f extends to a meromorphic function on ∆ 1 .
Proof.-We first prove that for every sequence (x n ), x n ∈ U , with lim n→∞ x n = p ∈ ∂U , one has lim n→∞ |f (x n )| = ∞ ∈ P 1 (C).Assume the contrary, i.e. there is a sequence and note their corresponding points in M as q 1 and q 2 .Then q 1 = q 2 .The sequences (x n , t 1 ) ∈ ∆ 1 × C and (x n , t 1 + f (x n )) ∈ ∆ 2 × C define the same sequence in M having q 1 and q 2 as accumulation points.So M is not Hausdorff, a contradiction.
In particular we have proved that the zeros of f do not accumulate to ∂U in ∆ 1 .So there is an open neighborhood V of ∂U in ∆ 1 such that the restriction of f to W := U ∩ V does not vanish.Let g := 1/f on W . Then g extends to a continuous function on V taking the value zero outside of U .The theorem of Rado implies that this function is holomorphic on V .It follows that the boundary ∂U is discrete in ∆ 1 and that f has a pole in each of the points of this set, so f is a meromorphic function on ∆ 1 .
Theorem 4.4.-The orbit space X * /C is Hausdorff.Consequently, C acts properly on X * .
Proof.-By virtue of the above lemma, in a neighborhood of two non-separable Corbits X is isomorphic to a domain in one of the model Stein surfaces discussed in the previous subsection.Since we have seen there that these surfaces are never globalizations, we arrive at a contradiction.Hence, all C-orbits are separable.

Examples
In this section we discuss several examples which illustrate our results.
5.1.Hyperbolic Stein surfaces with proper R-actions.-Let R be a compact Riemann surface of genus g ≥ 2. It follows that the universal covering of R is given by the unit disc ∆ ⊂ C and hence that R is hyperbolic.The fundamental group π 1 (R) of R contains a normal subgroup N such that π 1 (R)/N ∼ = Z.Let R → R denote the corresponding normal covering.Then R is a hyperbolic Riemann surface with a holomorphic Z-action such that R/Z = R.Note that Z is not contained in a one parameter group of automorphisms of R.
We have two mappings The map p : X → ∆ \ {0} is a holomorphic fiber bundle with fiber R. Since the Serre problem has a positive answer if the fiber is a non-compact Riemann surface ( [Mok82]), the suspension X = H × Z R is a hyperbolic Stein surface.The group R acts on H × R by t • (z, x) = (z + t, x) and this action commutes with the diagonal action of Z. Consequently, we obtain an action of R on X.
Lemma 5.1.-The universal globalization of the local C-action on X is given by which extends the R-action on X.We will show that X is orbit-connected in X * : Since [z + t, x] lies in X if and only if there exist elements In order to show that C acts properly on X * it is sufficient to show that C × Z acts properly on C × R. Hence, we choose sequences holds.Since Z acts properly on R, it follows that {m n } has a convergent subsequence, which in turn implies that {t n } has a convergent subsequence.Hence, the lemma is proven.
Proposition 5.2.-The quotient X/Z ∼ = ∆ * × R is not holomorphically separable and in particular not Stein.The quotient X * /C is biholomorphically equivalent to R/Z = R.
Proof.-It is sufficient to note that the map Φ : Proposition 5.3.-The quotient X/Z ∼ = ∆ * × R is not holomorphically separable and in particular not Stein.
Thus we have found an example for a hyperbolic Stein surface X endowed with a proper R-action such that the associated Z-quotient is not holomorphically separable.Moreover, the R-action on X extends to a proper C-action on a Stein manifold X * containing X as an orbit-connected domain such that X * /C is any given compact Riemann surface of genus g ≥ 2.

Counterexamples with domains in C
n .-There is a bounded Reinhardt domain D in C 2 endowed with a holomorphic action of Z such that D/Z is not Stein.However, this Z-action does not extend to an R-action.We give quickly the construction.
Let λ := 1 2 (3 + √ 5) and It is obvious that D is a bounded Reinhardt domain in C 2 avoiding the coordinate hyperplanes.The holomorphic automorphism group of D is a semidirect product Γ ⋉ (S 1 ) 2 , where the group Γ ≃ Z is generated by the automorphism (x, y) → (x 3 y −1 , x) and (S 1 ) 2 is the rotation group.Therefore the group Γ is not contained in a one-parameter group.Furthermore the quotient D/Γ is the (non-Stein) complement of the singular point in a 2-dimensional normal complex Stein space, a so-called "cusp singularity".These singularities are intensively studied in connection with Hilbert modular surfaces and Inoue-Hirzebruch surfaces, see e.g.[vdG88] and [Zaf01].
In the rest of this subsection we give an example of a hyperbolic domain of holomorphy in a 3-dimensional Stein solvmanifold endowed with a proper R-action such that the Z-quotient is not Stein.While this domain is not simply-connected, its fundamental group is much simpler than the fundamental groups of our two-dimensional examples.
Let G := ; a, b, c ∈ C be the complex Heisenberg group and let us consider its discrete subgroup Proposition 5.4.-The group Γ acts properly and freely on C 2 , and the quotient manifold C 2 /Γ is holomorphically separable but not Stein. Proof which shows that Γ acts properly and freely on C 2 .Moreover, we obtain the commutative diagram The group C * acts by multiplication in the first factor on C * × C * and this action commutes with the Z-action.One checks directly that the joint (C * × Z)-action on C * × C * is proper which implies that the map Y → T is a C * -principal bundle.Conseqently, Y is not Stein.
In order to show that Y is holomorphically separable, note that by [Oel92] this C * -principal bundle Y → T extends to a line bundle p : L → T with first Chern class c 1 (L) = −1.Therefore the zero section of p : L → T can be blown down and we obtain a singular normal Stein space Y = Y ∪ {y 0 } where y 0 = Sing(Y ) is the blown down zero section.Thus Y is holomorphically separable.
Let us now choose a neighborhood of the singularity y 0 ∈ Y biholomorphic to the unit ball and let U be its inverse image in C 2 .It follows that U is a hyperbolic domain with smooth strictly Levi-convex boundary in C 2 and in particular Stein.In order to obtain a proper action of R we form the suspension D = H × Γ U where Γ acts on H × U by (t, z, w) Hence, by virtue of [DG60] we only have to show that G/Γ is Stein.
For this we note first that G is a closed subgroup of SL(2, C) ⋉ C 2 which implies that G/Γ is a closed complex submanifold of X := SL(2, C) ⋉ C 2 /Γ.By [Oel92] the manifold X is holomorphically separable, hence G/Γ is holomorphically separable.Since G is solvable, a result of Huckleberry and Oeljeklaus ([HO86]) yields the Steinness of G/Γ.
One checks directly that the action of R × Γ on H × U is proper which implies that R acts properly on H × Γ U .

Steinness of
We summarize our remarks in the following Theorem 6.3.-Let D be a simply-connected bounded domain of holomorphy in C 2 admitting a non-compact connected identity component of its automorphism group.Then D is biholomorphic to a domain of the form where the functions u ′ , −o ′ are subharmonic in S.

3. 1 .
Locally proper actions.-Recall that the action of a Lie group G on a manifold M is called locally proper if every point in M admits a G-invariant open neighborhood on which the G acts properly.Lemma 3.1.-Let G × M → M be locally proper.
2. Without loss of generality we may assume that p 1 ∈ D and ζ • p 2 = (0, −2ζi, −i) ∈ D for some ζ ∈ C. We will show that the orbits R•p 1 and R•(ζ •p 2 ) cannot be separated by R-invariant open neighborhoods.

6. 1 .
Proper R-actions on D. -Let D ⊂ C n be a bounded domain and let Aut(D) 0 be the connected component of the identity in Aut(D).
Lemma 6.1.-A proper R-action by holomorphic transformations on D exists if and only if the group Aut(D) 0 is non-compact.
D/Z. -Now we give the proof of our main result.Theorem 6.2.-Let D be a simply-connected bounded domain of holomorphy in C 2 .Suppose that the group R acts properly by holomorphic transformations on D. Then the complex manifold D/Z is biholomorphically equivalent to a domain of holomorphy in C 2 .Proof.-Let D ⊂ C 2 be a simply-connected bounded domain of holomorphy.Since the Serre problem is solvable if the fiber is D, see[Siu76], the universal globalization D * is a simply-connected Stein surface,[CTIT00].Moreover, we have shown in Theorem 4.4, that C acts properly on D * .Since the Riemann surface D * /C is also simply-connected, it must be ∆, C or P 1 (C).In all three cases the bundle D * → D * /C is holomorphically trivial.So we can exclude the case that D * /C is compact and it follows thatD/Z ∼ = C * × (D * /C) is a Stein domain in C 2 .6.3.A normal form for domains with non-compactAut(D) 0 .-Let D ⊂ C 2 be a simply-connected bounded domain of holomorphy such that the identity component of its automorphism group is non-compact.As we have seen, this yields a proper R-action on D by holomorphic transformations and the universal globalization of the induced local C-action on D is isomorphic to C × S where S is either ∆ or C and where C acts by translation in the first factor.Moreover, there are plurisubharmonic functions u, −o : C × S → R ∪ {−∞} which fulfill u t • (z 1 , z 2 ) = u(z 1 , z 2 ) − Im(t) and o t • (z 1 , z 2 ) = o(z 1 , z 2 ) − Im(t)