The Minimum Principle

from a Hamiltonian Point of View

Peter Heinzner*

Received: February 24, 1998

Communicated by Thomas Peternell

Abstract. Let G be a complex Lie group and G_{ℝ} a real form of G. For a G_{ℝ}-stable domain of holomorphy X in a complex G-manifold we consider the question under which conditions the extended domain G · X is a domain of holomorphy. We give an answer in term of G_{ℝ}-invariant strictly plurisubharmonic functions on X and the associate Marsden-Weinstein reduced space which is given by the Kaehler form and the moment map associated with the given strictly plurisubharmonic function. Our main application is a proof of the so called extended future tube conjecture which asserts that G · X is a domain of holomorphy in the case where X is the N-fold product of the tube domain in ℂ^{4} over the positive light cone and G is the connected complex Lorentz group acting diagonally.

1991 Mathematics Subject Classification: Primary 32M05

Let G_{ℝ} be a connected real form of a complex Lie group G and X a G_{ℝ}-stable domain in a complex G-manifold Z such that G·X = Z. In this paper we consider the following question. Under which conditions on X is Z the natural domain of definition of the G_{ℝ}-invariant holomorphic functions on X? If Z is an open submanifold of a Stein manifold, then there is an envelope of holomorphy for Z. Consequently, every G_{ℝ}- invariant holomorphic function on X which extends to Z also extends to the envelope of holomorphy of Z. Thus one also has to ask under which additional requirements is Z a Stein manifold.

In order that an invariant holomorphic function extends to Z = G · X it is sufficient that X is orbit connected, i.e., for every z ∈ Z the set {g ∈ G; g · z ∈ X} is connected (see [H]). Thus under this condition the main question is whether Z is a Stein manifold. Now if Z is a domain in a Stein manifold V, then Z itself is a Stein manifold if one can find a plurisubharmonic function Ψ on Z which goes to +∞ at every boundary point of ∂Z ⊂ V. There is a natural way to construct

* Supported by a Heisenberg Stipendium of the Deutsche Forschungsgemeinschaft G-invariant plurisubharmonic functions out of G_{ℝ}-invariant functions on X which was first proposed by Loeb in [L]. In this paper Loeb used an extended version of Kiselman’s minimum principle ([K]) in order to construct invariant plurisubharmonic functions. The main idea is the following. Assume that there is a nice quotient π : Z → Z/G and let ϕ be a smooth G_{ℝ}-invariant plurisubharmonic function on X which is a strictly plurisubharmonic exhaustion on each fibre of π|X. Then the fibre wise minimum of ϕ defines a function ψ on Z/G which is a candidate for a plurisubharmonic function on Z/G. This procedure can be described in terms of Hamiltonian actions as follows.

Documenta Mathematica 3 (1998) 1–14

Assume for simplicity that ϕ is strictly plurisubharmonic. Then ω := 2i∂ ¯∂ϕ defines an invariant Käher form on X and μ(x)(ξ) = dϕ(Jξ_{X}) is the associated moment map μ : X → 𝔤_{ℝ}^{∗}. In this situation μ^{−1}(0) is the set of fibre wise critical points of ϕ which in good cases are exactly the points such that the restriction of ϕ to the fibre attains its minimum. Again under some additional assumption, it then follows from the principle of symplectic reduction that the reduced space μ^{−1}(0)/G_{ℝ} has a symplectic structure which in fact is Kählerian and moreover is given by the function ψ which is induced on μ^{−1}(0)/G_{ℝ} by ϕ|μ^{−1}(0). It turns out that in the situation under consideration the procedures given by symplectic reduction and minimum principle are compatible. This is well known in the case where G_{ℝ} is a compact Lie group (see e.g. [H-H-L], where a much more general result is proved) and we give here precise conditions such that it also works for a non compact group G_{ℝ}.

The application of Loeb’s Minimum Principle is limited mainly to the case of free G_{ℝ}-actions. For the more general case of proper actions it seems that the Hamiltonian point of view is much more adequate. Moreover, for applications it is necessary to consider also domains X of G-spaces Z which do not admit a geometrical quotient Z/G. A typical example is given by the so called extended future tube which we will describe next.

Let < , > denote the Lorentz product on ℝ^{4} and also its ℂ-bilinear extension to ℂ^{4}. The future tube T is by definition the tube domain in ℂ^{4} = ℝ^{4} + iℝ^{4} over the positive light cone C^{+} = {y = (y_{0}, y_{1}, y_{2}, y_{3}) ∈ ℝ^{4}; y_{0} > 0, < y, y >= (y_{0})^{2} − (y_{1})^{2} − (y_{2})^{2} − (y_{3})^{2} > 0}, i.e.,

T = {z ∈ ℂ^{4}; Im z ∈ C^{+}} .

This domain is invariant under the action of the connected component G_{ℝ} of the identity of the homogeneous Lorentz group O_{ℝ}(1, 3). Now consider the N-fold product T^{N} with the diagonal action of G_{ℝ}. The extended future tube (T^{N})^{ℂ} is by definition the orbit of T^{N} under the action of the complexified group G of G_{ℝ}. In other words

(T^{N})^{ℂ} = G · T^{N} = {(g · z_{1}, …, g · z_{N}); g ∈ G, z_{j} ∈ T } .

Note that G is the group SO_{4}(ℂ) which is defined by the quadratic form < , >. Although there is no geometric quotient of Z, we have a quotient π : (ℂ^{4})^{N} → (ℂ^{4})^{N}//G which is given by the invariant holomorphic functions on (ℂ^{4})^{N} and it is a fundamental fact that the extended tube (T^{N})^{ℂ} is saturated with respect to π ([H-W], see §3 for additional remarks). In this case it turns out that this invariant theoretical quotient has sufficiently many good properties in order to apply the main result of this paper which we formulate now.

Let V be a Stein G-manifold such that there exists almost a quotient π : V → V//G. More precisely we will assume that V//G is a complex space, π : V → V//G is a G-invariant surjective holomorphic map and for an analytically Zariski open πsaturated subset V^{0} of V the restriction map π : V^{0} → V^{0}//G is a holomorphic fibre bundle with typical fibre G/H. Thus V^{0}//G = V^{0}/G is a geometric quotient. Let X be a G_{ℝ}-stable domain in V such that Z := G · X is saturated with respect to π : V → V//G.

Theorem 1. Let ϕ : X → ℝ be a smooth non-negative G_{ℝ}-invariant plurisubharmonic function and assume that

(i) The fibres of π restricted to X^{0} := X ∩ V^{0} are connected, (ii) the restriction of ϕ^{0} := ϕ|X^{0} to the fibres of π restricted to X^{0} is strictly

plurisubharmonic, (iii) ϕ^{0} is proper mod G_{ℝ} along π|Z^{0} where Z^{0} := V^{0} ∩ Z and (iv) ϕ is a weak exhaustion of X over V//G, Then Z = G · X is a Stein manifold.

In the case where G_{ℝ} acts properly on X^{0} condition (iii) means that the map ϕ^{0} × π|X^{0} : X^{0} → ℝ × (Z^{0}//G) induces a proper map X^{0}/G_{ℝ} → ℝ × (Z^{0}//G). By a weak exhaustion of X over V//G we mean a function which goes to +∞ on a sequence if the corresponding sequence in V//G converges to a boundary point of Z//G in V//G.

In the case where the G-action on Z^{0} is assumed to be free, the theorem can be proved rather directly by applying Loeb’s minimum principle. For a compact group it is a consequence of the methods presented in [H-H-K] (see also [H-H-L]).

In the last section we recall some previously known facts proved in [H-W] together with a more recent result in [Z] about the orbit geometry of the extended future tube in order to verify that the conditions of Theorem 1 are satisfied in the case of the extended future tube. This leads to a conceptual proof of the so called extended future tube conjecture in the last section.

Theorem 2. The extended future tube is a domain of holomorphy.

This result has conjecturally been known in constructive quantum field theory for more then thirty years. For its relevance and other publications concerning problems related to it we refer the reader to the literature ([B-L-T], [H-S], [J], [S-W], [S-V]).

There is a proof of Theorem 2 in [Z] which due to several mistakes and gaps is difficult to understand.

1. Hamiltonian actions on Kähler spaces.

Let G_{ℝ} be a real connected Lie group and X a complex G_{ℝ}-space, i.e., G_{ℝ} acts on X by holomorphic transformations such that the action G_{ℝ} × X → X, (g, x) → g · x, is real analytic. If ω is a smooth G_{ℝ}-invariant Kähler structure on X, then a G_{ℝ}- equivariant smooth map μ from X into the dual 𝔤_{ℝ}^{∗} of the Lie algebra 𝔤_{ℝ} of G_{ℝ} is said to be an equivariant moment map if

dμ_{ξ} = ı_{ξ_{X}} ω

holds on every G_{ℝ}-stable complex submanifold Y of X. Here ω denotes the Kähler form on Y induced by the Kählerian structure on X (see [H-H-L]), μ_{ξ} :=< μ, ξ > is the component of μ in the direction of ξ ∈ 𝔤_{ℝ}, ξ_{X} is the vector field on X induced by ξ and ι_{ξ_{X}} ω denotes the one form given by contraction, i.e., η → ω(ξ, η).

Example. If ω is given by a smooth strictly plurisubharmonic G_{ℝ}-invariant function ϕ, i.e., ω = 2i∂ ¯∂ϕ on every smooth part of X, then

μ_{ξ}(x) := dϕ(Jξ_{X}) = (i(∂ − ¯∂)ϕ)(ξ_{X}) = d^{c}ϕ(ξ_{X})

defines an equivariant moment map. This follows from invariance of ϕ, since in this case we have

dμ_{ξ} = dı_{ξ_{X}} d^{c}ϕ = −ı_{ξ_{X}} dd^{c}ϕ = ı_{ξ_{X}} 2i∂ ¯∂ϕ .

Here we use the formula

L_{ξ}α = ı_{ξ}dα + dı_{ξ}α

for all vector fields ξ and differential forms α where L_{ξ} denotes the Lie derivative in the direction of ξ.

Later we will need the following fact about the zero level set of μ.

Lemma. Assume that X is smooth and that G_{ℝ} acts properly on X. If the dimension of the G_{ℝ}-orbits in μ^{−1}(0) is constant, then μ^{−1}(0) is a submanifold of X.

Proof. Since the action is assumed to be proper, there is a local normal form for the moment map (see e.g. [A] or [H-L]). The statement is an easy consequence of this fact (see e.g. [A]. In [S-L] the argument is given for a compact group G_{ℝ}). □

Remark 1. It can be shown that the converse of the Lemma also holds. We will not use this fact here.

Remark 2. The properness assumption is very often satisfied. Since one may assume that G_{ℝ} acts effectively, G_{ℝ} is a Lie subgroup of the group I of isometries of the Riemannian manifold X. The group of isometries acts properly on X and consequently the G_{ℝ}-action on X is proper if and only if G_{ℝ} is a closed subgroup of I. This is the case if and only if there is a point x ∈ X such that G_{ℝ} · x is closed and the isotropy group (G_{ℝ})_{x} := {g ∈ G_{ℝ}; g · x = x} is compact.

Remark 3. If G_{ℝ} acts such that the isotropy groups are discrete, then μ has maximal rank. Thus in this case μ^{−1}(0) is obviously a submanifold of X. Moreover T_{x}(μ^{−1}(0)) = ker dμ(x) for all x ∈ μ^{−1}(0).

2. Hamiltonian actions on invariant domains

Let G be a connected complex Lie group and Z a holomorphic G-space, i.e., the action G × Z → Z is assumed to be a holomorphic map. Let G_{ℝ} be a connected real form of G. By an invariant domain in Z we mean in the following a G_{ℝ}-stable connected open subspace X of Z. In the homogeneous case we have the following Lemma 1. Let X be an invariant domain in Z and assume that Z is G-homogeneous. If the zero level set of μ : X → 𝔤_{ℝ} is not empty, then μ^{−1}(0) is a Lagrangian submanifold of X and each connected component of μ^{−1}(0) is a G_{ℝ}-orbit.

Proof. For z_{0} ∈ X let N be an open convex neighborhood of 0 ∈ 𝔤_{ℝ} such that U := G_{ℝ} · exp iN · z_{0} ⊂ X. Since G_{ℝ} · exp iN is a neighborhood of G_{ℝ} in G, the set U is a neighborhood of G_{ℝ} · z_{0} in X. The proof of Lemma 1 is a consequence of the following

Claim. U ∩ μ^{−1}(0) = G_{ℝ} · z_{0} for z_{0} ∈ μ^{−1}(0).

In order to proof the claim, let z ∈ U ∩ μ^{−1}(0) be given. Then there are h ∈ G_{ℝ} and ξ ∈ N such that z = h exp iξ · z_{0} ∈ U ∩ μ^{−1}(0). Thus exp iξ · z_{0} ∈ μ^{−1}(0) ∩ U and z_{t} := exp itξ ·z_{0} ∈ U for t ∈ [0, 1]. Note that Jξ_{X}(x) = \frac{d}{dt}|_{t=0} exp itξ ·x is the gradient flow of μ_{ξ} with respect to the Riemannian metric induced by ω. Thus, if z_{t} is not constant, then t → μ_{ξ}(z_{t}) is strictly increasing. This contradicts μ_{ξ}(z_{0}) = 0 = μ_{ξ}(z_{1}). Therefore z_{0} = exp itξ · z_{0} for all t ∈ ℝ. This implies z = h · z_{1} = h · z_{0} ∈ G_{ℝ} · z_{0}.

It is a consequence of the claim that every G_{ℝ}-orbit is closed in X. Therefore every component of μ^{−1}(0) is a G_{ℝ}-orbit. It remains to show that these orbits are Lagrangian. Since μ(G_{ℝ} · z_{0}) = 0 we have

0 = dμ_{ξ}(η_{X}(z_{0})) = ω(ξ_{X}(z_{0}), η_{X}(z_{0}))

for all ξ, η ∈ 𝔤_{ℝ}. This means that G_{ℝ} · z_{0} is an isotropic submanifold of X. In particular, dim_{ℝ} G_{ℝ} · z_{0} ≤ dim_{ℂ} X. In general the tangent space T_{z_{0}}(G_{ℝ} · z_{0}) spans T_{z_{0}}X over ℂ. Thus dim_{ℝ} G_{ℝ} · z_{0} ≥ dim_{ℂ} G · z_{0} = dim_{ℂ} X. This shows that dim_{ℝ} G_{ℝ} · z_{0} = \frac{1}{2} dim_{ℝ} X. Hence G_{ℝ} · z_{0} is Lagrangian. □

Every Lagrangian submanifold of a Kähler manifold is totally real. Thus, if Z is G-homogeneous, then μ^{−1}(0) is a totally real submanifold of X. Note that the G_{ℝ}-orbits in μ^{−1}(0) are closed since they are connected components of the zero fibre of μ. Now if G_{ℝ} is such that 0 ∈ 𝔤_{ℝ}^{∗} is the only G_{ℝ}-fixed point, then x ∈ μ^{−1}(0) if and only if the orbit G_{ℝ} · x is isotropic. This condition holds for example for a semisimple Lie group.

It almost never happens that there is a G_{ℝ}-invariant Kähler form ω which is defined on Z. For example, if G_{ℝ} is a simple non compact Lie group or more generally a semisimple Lie group without compact factors, then there does not exist a G_{ℝ}- invariant Kähler form on a non trivial holomorphic G-manifold Z. In order to see this, recall that since G_{ℝ} is semisimple there is a moment map μ : Z → 𝔤_{ℝ}^{∗}. Now let 𝔤_{ℝ} = 𝔨 ⊕ 𝔭 be a Cartan decomposition where 𝔨 is the Lie algebra of the maximal compact subgroup of G_{ℝ}. Then 𝔲 = 𝔨 ⊕ i𝔭 is the Lie algebra of the maximal compact subgroup U of G. For ξ ∈ i𝔭 the image of the one-parameter group γ : t → exp itξ lies in U and therefore there is a basis of 𝔭 consisting of ξ’s such that the image of γ is compact, i.e., isomorphic to S^{1}. But γ is the flow of the gradient vector field of μ_{ξ} and therefore t → μ_{ξ}(γ(t) · z) is strictly increasing for every z ∈ Z. This implies that γ acts trivially on Z. Since G is semisimple and contains no compact factor, G itself is the smallest complex subgroup of G which contains exp 𝔭. Thus G acts trivially on Z.

A geometric interpretation of the zero fibre μ^{−1}(0) of an equivariant moment map μ : X → 𝔤_{ℝ}^{∗} associated to a smooth G_{ℝ}-invariant strictly plurisubharmonic function ϕ : X → ℝ (see Section 1, Example) can be given in the case where X is an invariant domain in Z as follows. For x ∈ X let Ω(x) := {g · x; g ∈ G and g · x ∈ X} be the local G-orbit of G through x in X where (g, x) → g · x denotes the G-action on Z. Then by G_{ℝ}-invariance of ϕ we have

μ^{−1}(0) = {x ∈ X; x is a critical point of ϕ|Ω(x)} .

We consider now invariant domains X in G-homogeneous spaces Z such that there is a moment map associated to ϕ : X → ℝ more closely. In order to do that we first introduce the notion of an exhaustion mod G_{ℝ}.

Let F be a complex space with a proper G_{ℝ}-action and let F/G_{ℝ} be the space of G_{ℝ}-orbits endowed with the quotient topology. A G_{ℝ}-invariant function f : F → ℝ is said to be proper mod G_{ℝ} if the induced map f̄: F/G_{ℝ} → ℝ is proper. The map f is said to be an exhaustion mod G_{ℝ} if f̄ is an exhaustion, i.e., if for all r ∈ ℝ the set {q ∈ F/G_{ℝ}; f̄(q) < r} is relatively compact in F. Note that a G_{ℝ}-invariant continuous function which is bounded from below is proper mod G_{ℝ} if and only if it is an exhaustion mod G_{ℝ}.

Lemma 2. Let Z be G-homogeneous and assume that the G_{ℝ}-action on X is proper. Let ϕ : X → ℝ be a smooth strictly plurisubharmonic G_{ℝ}-invariant function which is an exhaustion mod G_{ℝ}. Then there is a z_{0} ∈ X such that

G_{ℝ} · z_{0} = μ^{−1}(0) = {z ∈ X; ϕ(z) is a minimal value of ϕ} .

Proof. Since ϕ is plurisubharmonic and an exhaustion mod G_{ℝ} there is a point z_{0} ∈ X which is a minimum for ϕ. In particular, μ^{−1}(0) is not empty where μ denotes the moment map associated with ϕ. We have to prove that μ^{−1}(0) is connected. By Lemma 1, every connected component of the set M_{ϕ} = μ^{−1}(0) of critical points of ϕ is a G_{ℝ}-orbit. We claim that the G_{ℝ}-orbits are non degenerate in the sense that the Hessian of ϕ in normal directions is positive definite. This is seen as follows.

The vector fields Jξ_{X}, ξ ∈ 𝔤_{ℝ} span the normal space at x ∈ M_{ϕ} and

(Jξ_{X})(Jξ_{X}(ϕ)) = ı_{Jξ_{X}} dμ_{ξ} = ω(ξ_{X}, Jξ_{X}) .

Hence the Hessian at x ∈ M_{ϕ} is positive in the normal directions. Since ϕ is proper mod G_{ℝ} and the gradient vector field of ϕ with respect to the G_{ℝ}-invariant Kähler metric given by 2i∂ ¯∂ϕ is G_{ℝ}-invariant, Lemma 2 follows from standard arguments in Morse Theory. □

In the situation of Lemma 2 every critical point of ϕ is a minimum and the set of these points is a G_{ℝ}-orbit and coincides with μ^{−1}(0).

We will now generalize the results in the homogeneous case to spaces Z which possess a geometric G-quotient and X is a weakly orbit connected invariant domain in Z. Here a G_{ℝ}-stable subset X of Z is said to be weakly orbit connected if for every x ∈ X the local G-orbit Ω(x) := {g · x ∈ X; g ∈ G} is connected.

Remark 1. A G_{ℝ}-invariant set X in Z is said to be orbit connected if for every x ∈ X the set Ω_{x} := {g ∈ G; g · x ∈ X} is connected. This is a stronger concept then weakly orbit connectedness.

Let Z be a holomorphic G-space such that there is a geometric quotient π : Z → Z/G. By this we mean that the orbit space Z/G is a complex space such that the quotient map π : Z → Z/G is holomorphic. Moreover we assume that the structure sheaf of Z/G is the sheaf of invariants, i.e., for an open subset Q of Z/G a function f : Q → ℂ is holomorphic if and only if f ∘ π : π^{−1}(Q) → ℂ is holomorphic.

Now let X ⊂ Z be an invariant domain which lies surjectively over Z/G or equivalently such that Z = G · X. Assume that G_{ℝ} acts properly on X and that X is weakly orbit connected. Let ϕ : X → ℝ be a smooth G_{ℝ}-invariant strictly plurisubharmonic function which is an exhaustion mod G_{ℝ} along π, i.e., π^{−1}(C)∩{x ∈ X; ϕ(x) ≤ r}/G_{ℝ} ⊂ X/G_{ℝ} is compact for every compact subset C in Z/G and r ∈ ℝ. We set M_{ϕ} = μ^{−1}(0) where μ : X → 𝔤_{ℝ} denotes the moment map associated with ϕ.

Proposition 1. The map ī: M_{ϕ}/G_{ℝ} → Z/G induced by the inclusion ı : M_{ϕ} → Z is a homeomorphism. If X is a manifold, then M_{ϕ} is smooth and

T_{x}M_{ϕ} = ker dμ(x)

holds for all x ∈ M_{ϕ}.

Proof. The map ī is continuous and by Lemma 2 it is also a bijection. We claim that ī is proper. Since the G_{ℝ}-action on M_{ϕ} is proper, M_{ϕ}/G_{ℝ} is a locally compact topological space. Thus properness of ī implies that ī is a homeomorphism.

Let (q_{n}) be a sequence in M_{ϕ}/G_{ℝ} and x_{n} a point in M_{ϕ} which lies over q_{n}. Assume that (π(x_{n})) = (ī(q_{n})) has a limit in Z/G and let x_{0} ∈ M_{ϕ} be a point which lies over lim π(x_{n}). If some subsequence of ϕ(x_{n}) goes to infinity, then we may assume ϕ(x_{n}) > ϕ(x_{0}) + 1 for all n. Since π : Z → Z/G is an open map, there are g_{n} ∈ G such that lim g_{n} · x_{n} = x_{0} for some subsequence. This is a contradiction since ϕ(x_{n}) < ϕ(g_{n} · x_{n}) for all n such that g_{n} · x_{n} ∈ X. Thus, since ϕ is assumed to be an exhaustion mod G_{ℝ} along π, there are h_{n} ∈ G_{ℝ} such that a subsequence of (h_{n} · x_{n}) converges to x_{0}. This implies that a subsequence of (q_{n}) converges in M_{ϕ}. So far we proved that ī is a homeomorphism.

Assume now that X is smooth. The existence of a geometric quotient implies that the dimension of the G-orbits in Z is constant and therefore this is also true for the G_{ℝ}-orbits in M_{ϕ} (Lemma 1). Thus M_{ϕ} is a submanifold of X (Section 1, Lemma). Since T_{x}M_{ϕ} is a subspace of ker dμ(x) and ker dμ(x) = T_{x}(G_{ℝ} · x) + T_{x}(G · x)^{⊥}, the claim follows from the obvious dimension count as follows. Let d := dim_{ℝ} G_{ℝ} · x for x ∈ M_{ϕ}. Note that d is the complex dimension of the π-fibres. Thus dim_{ℝ} M_{ϕ} = dim_{ℝ} M_{ϕ}/G_{ℝ} + d = dim_{ℝ} Z/G + d = dim_{ℝ} T_{x}(G · x)^{⊥} + dim_{ℝ} G_{ℝ} · x implies that T_{x}M_{ϕ} = ker dμ(x) for all x ∈ M_{ϕ}. □

Remark 2. Without a reference to an embedding into a holomorphic G-space one can show that μ^{−1}(0)/G_{ℝ} is a complex space in a natural way (see [A-H-H] and [A]).

If G_{ℝ} does not act properly on X, then let \overline{G}_{ℝ} be the closure of G_{ℝ} in the group I of isometries of the Kähler manifold X. Since the G_{ℝ}-orbits in M_{ϕ} = μ^{−1}(0) are closed (Lemma 1), it follows that they coincide with the \overline{G}_{ℝ}-orbits. Moreover ϕ is \overline{G}_{ℝ}-invariant and M_{ϕ} = μ̄^{−1}(0) =: \overline{M}_{ϕ}, where μ̄ is the moment map associated with ϕ. Now if one redefines an exhaustion mod G_{ℝ} along π in terms of sequences in X, then also in this case M_{ϕ} is smooth and T_{x}M_{ϕ} = T_{x}(G_{ℝ} · x) ⊕ T_{x}(G · x)^{⊥} = ker dμ(x) holds for all x ∈ M_{ϕ}.

Proposition 1 can be generalized to the case where ϕ : X → ℝ is only assumed to be plurisubharmonic and strictly plurisubharmonic on the fibres. More precisely we have the following consequence which can be thought of as a version of Loeb’s minimum principle (see [L]).

Corollary 1. Let X ⊂ Z be a weakly orbit connected invariant domain with π(X) = Z and ϕ : X → ℝ a smooth G_{ℝ}-invariant plurisubharmonic function which is an exhaustion mod G_{ℝ} along π such that the restriction of ϕ to the local G-orbits in X is a strictly plurisubharmonic exhaustion mod G_{ℝ}. If π : Z → Z/G is a holomorphic bundle, then

(i) M_{ϕ} = μ^{−1}(0) is smooth where μ : X → 𝔤_{ℝ}^{∗}, μ_{ξ} = dϕ(Jξ_{X}), (ii) T_{x}(M_{ϕ}) = ker dμ(x) for all x ∈ M_{ϕ}. (iii) M_{ϕ}/\overline{G}_{ℝ} is homeomorphic to Z/G and the function ψ : Z/G → ℝ which is

induced by ϕ|M_{ϕ} is a smooth plurisubharmonic function.

Proof. We may assume that G_{ℝ} acts properly on X and, since the statements are local over Z/G that Z/G is a Stein manifold. Let ρ : Z → ℝ be the the pull back of a strictly plurisubharmonic function on Z/G. Then ϕ + ρ is G_{ℝ}-invariant, strictly plurisubharmonic and an exhaustion mod G_{ℝ} on the local G-orbits in X. Since dρ(Jξ_{X}) = 0 for all ξ ∈ 𝔤_{ℝ}, the moment map associated with ϕ + ρ is the same as the moment map associated with ϕ. Thus Proposition 1 implies directly (i), (ii) and the first part of (iii). It remains to show that ψ : Z/G → ℝ is a smooth plurisubharmonic function.

For the plurisubharmonicity of ψ we recall the calculation in [H-H-L], §2. For z ∈ M_{ϕ} we have T_{z}(M_{ϕ}) = ker dμ(z) = T_{z}(G_{ℝ} · z) ⊕ T_{z}(G · z)^{⊥}. We may assume that Z = G/H × Δ where Δ is an open neighborhood of 0 in ℂ^{d} ≅ T_{z}(G · z)^{⊥}, and π(z) = 0 where π is given by the projection on the second factor. Furthermore there is a section η : Δ → M_{ϕ}, η(w) = (σ(w), w) and therefore we have ψ(w) = ϕ(η(w)). A direct calculation shows that

∂ ¯∂ψ(0) = ∂ ¯∂ϕ(η(0)) .

Here one has to use that dϕ(z) = 0 and that dσ(0) = 0. Thus ψ is plurisubharmonic and smooth. □

If ϕ is strictly plurisubharmonic, then the proof shows that ψ is also strictly plurisubharmonic. For a proper G_{ℝ}-action the space Z/G is then given by symplectic reduction M_{ϕ}/G_{ℝ} and the induced Kählerian structure on Z/G is determined by the function ψ(q) = inf_{x∈π^{−1}(q)∩X} ϕ(x) which is obtained by applying the minimum principle ([L]). Thus symplectic reduction and the minimum principle are compatible procedures.

For the remainder of this section we assume now that Z is a holomorphic Gmanifold such that there is almost a quotient Z//G. More precisely we will assume that Z//G is a complex space, π : Z → Z//G is a surjective G-invariant holomorphic map and there is an analytically Zariski open π-saturated subset Z^{0} of Z such that π : Z^{0} → Z^{0}//G is a geometric quotient, i.e., Z^{0}//G = Z^{0}/G. Moreover, for the sake of simplicity we assume that π : Z^{0} → Z^{0}//G is a holomorphic fibre bundle.

Now let X be an invariant domain in Z with π(X) = Z and assume that X^{0} := X ∩Z^{0} is weakly orbit connected. Let ϕ be a G_{ℝ}-invariant plurisubharmonic function such that ϕ^{0} := ϕ|X^{0} is smooth, strictly plurisubharmonic on the local G-orbits in X^{0} and an exhaustion mod G_{ℝ} along π|Z^{0}. Thus the restriction ϕ^{0} := ϕ|M_{ϕ}^{0}, M_{ϕ}^{0} := M_{ϕ} ∩ Z^{0} induces a plurisubharmonic function ψ^{0} : Z^{0}//G → ℝ.

Lemma 3. There is a unique G-invariant plurisubharmonic function Ψ : Z → [−∞, +∞) which extends Ψ^{0} := ψ^{0} ∘ π|Z^{0}.

Proof. The function Ψ(z) = inf_{g∈Ω_{z}} ϕ(g · z) is upper semi-continuous on Z where Ω_{z} :=

{g ∈ G; g · z ∈ X}. Now Ψ = Ψ^{0} on Z^{0} (Lemma 2), and Z \ Z^{0} is a proper analytic subset of Z. Thus Ψ is plurisubharmonic and by definition G-invariant. □

Remark 3. If Z//G is smooth and π is an open map, then ψ^{0} extends uniquely to a plurisubharmonic function ψ on Z//G. Of course in this case we have ψ(q) = inf_{x∈F_{q}} ϕ(x),

where F_{q} := π^{−1}(q) ∩ X. If ϕ|F_{q} is an exhaustion mod G_{ℝ}, M_{ϕ} intersects every G_{ℝ}- stable closed analytic subset of F_{q} non trivially. But it might happen that M_{ϕ} ∩ F_{q} is a union of several G_{ℝ}-orbits. On the other hand for q ∈ Z^{0}//G the intersection is exactly one G_{ℝ}-orbit.

Assume now in addition that Z is an open G-stable subspace of a holomorphic Stein G-manifold V which is saturated with respect to π : V → V//G. We say that ϕ : X → ℝ is a weak exhaustion of X over V//G if lim sup ϕ(z_{n}) = +∞ for any sequence (z_{n}) in X such that (π(z_{n})) converges to some q_{0} in the boundary ∂(Z//G) in V//G.

Theorem. Let Z be a G-stable π-saturated open subspace of V, X an invariant domain in Z with G · X = Z and ϕ : X → ℝ a G_{ℝ}-invariant plurisubharmonic function. Assume that

(i) X^{0} is weakly orbit connected, (ii) the restriction of ϕ^{0} := ϕ|X^{0} to the local G-orbits is strictly plurisubharmonic, (iii) ϕ^{0} is an exhaustion mod G_{ℝ} along π|Z^{0} and (iv) ϕ is a weak exhaustion of X over V//G, Then Z = G · X is a Stein manifold.

Proof. Let z_{0} ∈ ∂Z and z_{n} ∈ Z be such that z_{0} = lim z_{n}. We have to show lim sup Ψ(z_{n}) = +∞. Thus assume that Ψ(z_{n}) < r for all n and some r ∈ ℝ.

There are w_{n} ∈ G · M_{ϕ}^{0} = Z^{0} such that Ψ(w_{n}) < r and z_{0} = lim w_{n}. Let w_{n} = g_{n} · x_{n} where g_{n} ∈ G and x_{n} ∈ M_{ϕ}^{0}. Now Ψ(w_{n}) = Ψ(x_{n}) = ϕ(x_{n}) < r and, since Z = G · X is saturated, π(x_{n}) = π(w_{n}) → π(z_{0}) ∈ ∂(Z//G). This contradicts the assumption that ϕ is a weak exhaustion. Thus Z is a domain in a Stein manifold with a plurisubharmonic weak exhaustion function and therefore Stein. □

Remark 4. Elementary examples show that for a Stein G_{ℝ}-manifold some conditions are necessary in order that G · X is a Stein manifold. For example there is an Sl_{2}(ℝ)- invariant domain Ω of holomorphy in ℂ^{2} such that Sl_{2}(ℂ) · Ω = ℂ^{2} \ {0}.

Now let G be complex reductive group and assume that the semistable quotient π : Z → Z//G exists (see [H-M-P]). Thus Z//G is a complex space whose structure sheaf O_{Z//G}(U) = O_{Z}(π^{−1}(U)^{G} is the sheaf of invariants and every point in Z//G has an open Stein neighborhood such that the inverse image in Z is Stein. For example, if V is a holomorphic Stein G-manifold, then a semistable quotient V//G alway exists. Moreover it is shown in [H-M-P] that Z is a Stein space if and only if Z//G is a Stein space.

Assume that Z is connected and that some orbit of maximal dimension is closed. Then there exists a proper analytic subset A in Z//G such that Z^{o}//G = Z//G \ A is a geometric quotient of Z^{o} := π^{−1}(Z//G \ A). In particular, every fibre of π|Z^{o} is G-homogeneous or equivalently the dimension of the G-orbits in Z^{o} is constant. Every x ∈ Z^{o} has a G-stable neighborhood U which is G-equivariantly biholomorphic to G×_{H} S where H is the isotropy group of G at x and S is a Stein space such that the connected component H^{0} of the identity of H acts trivially on S. Here G×_{H} S denotes the bundle associated to the H-principal bundle G → G/H. Thus locally Z^{o}//G is given by S/Γ where Γ := H/H_{0} is a finite group. Moreover, there is an analytically Zariski open G-stable subset Z^{oo} of Z which is contained in Z^{o} such that the isotropy type is constant. This implies that Z^{oo} is a fibre bundle over Z^{oo}//G ⊂ Z//G.

3. Orbit geometry of the future tube.

In the following it will be convenient to introduce a linear coordinate change such that < z, z >= (z_{0})^{2} − (z_{1})^{2} − (z_{2})^{2} − (z_{3})^{2} has the form z_{0}z_{1} − z_{2}z_{3}. Thus we set

Z := ({x \atop z} {y \atop w} ) = ({z_{0} + z_{3} \atop z_{1} + iz_{2}} {z_{1} − iz_{2} \atop z_{0} − z_{3}} )

and obtain det Z =< z, z > and det Im Z =< Im z, Im z > where Im Z := \frac{1}{2i}(Z − Z̄^{t}).

Let H := {Z ∈ V; Im Z > 0} denote the generalized upper half plane where V := ℂ^{2×2}. Note that H is just the tube over the positive light cone in the new coordinates. Moreover H is stable with respect to the action of G_{ℝ} := SL_{2}(ℂ) which is given by G_{ℝ} × H → H, (g, Z) → g ∗ Z := gZḡ^{t}. This action is not effective. The ineffectivity consists of Γ = {+I, −I} and the quotient SL_{2}(ℂ)/Γ is the connected component of the identity of the homogeneous Lorentz group.

Let H^{N} := H × ⋯ × H ⊂ V × ⋯ × V =: V^{N} denote the N-fold product of H and set G := (G_{ℝ})^{ℂ} = SL_{2}(ℂ) × SL_{2}(ℂ) where G_{ℝ} is embedded in G via g → (g, ḡ). The diagonal G_{ℝ} action on V^{N} extends to a holomorphic G action G × V^{N} → V^{N}, ((g, h), Z^{1}, …, Z^{N}) → (g, h) ∗ (Z^{1}, …, Z^{N}) := (gZ^{1}h^{t}, …, gZ^{N}h^{t}).

Theorem. The extended future tube (H^{N})^{ℂ} := G ∗ H^{N} is a domain of holomorphy.

In the proof we will make an axiomatic use of the following statements

Fact 1 (see Streater Wightman [S-W], p. 66). The set H^{N} is orbit connected in V^{N}, i.e., {g ∈ G; g ∗ Z ∈ H^{N}} is connected for every Z ∈ V^{N}.

Fact 2. The extended future tube G ∗ H^{N} is saturated with respect to π : V^{N} → V^{N}//G.

Fact 2 implies that the semistable quotient G ∗ H^{N}//G exists and is an open subset of V^{N}//G. The quotient map is given by restricting π : V^{N} → V^{N}//G to G ∗ H^{N}.

There does not seem to be a proof in the literature of Fact 2 but there is a detailed proof for the whole complex orthogonal group in [H-W]. A slight modification of the proof there can be used for a proof of Fact 2. In order to be complete let us recall briefly the main steps. First we note that it is sufficient to show the following (see e.g. [H]).

Claim. If Z ∈ H^{N}, then the unique closed orbit G ∗ W in the closure of \overline{G ∗ Z} lies in G ∗ H^{N}.

This can be seen as follows. Let < , > be the complex Lorenz product, i.e., the symmetric bilinear form on V which is associated to the quadratic form det : V → V. Thus V is just the standard representation of G̃ := O_{4}(ℂ). Note that G̃ has two connected components and the connected component of the identity is G. The functions (Z^{1}, …, Z^{N}) →< Z^{i}, Z^{j} >, form a set of generators for the algebra of the G̃-invariant polynomials on V^{N}. Thus the image of V^{N} in the set of symmetric N × N-matrices of the map π̃ which sends (Z^{1}, …, Z^{N}) to the matrix (< Z^{i}, Z^{j} >) is an affine variety which is isomorphic to V^{N}//G̃.

The matrices of rank 3 or 4 correspond to fibres of π̃ which are closed G̃-orbits. It follows that the G-orbit through every point Z ∈ H^{N} such that the rank r of Z̃ is greater or equal to 3 is already closed. Now assume that r ≤ 2. In this case the following is shown in [H-W]: There exists an g ∈ G̃, α_{j} ∈ ℂ and an ω ∈ V with < ω, ω >= 0 =< ω, W^{j} > such that

Z^{j} = g ∗ W^{j} + α_{j}ω , j = 1, …, N .

The proof actually shows that one can choose g ∈ G, i.e., det g = 1. Now an argument of Hall-Wightman ([H-W], p.21) implies that g∗W^{j} ∈ H for all j, i.e., G∗W ⊂ G∗H^{N}.

Fact 3. The function ϕ : H^{N} → ℝ, ϕ(Z^{1}, …, Z^{N}) := \frac{1}{det Im Z^{1}} + ⋯ + \frac{1}{det Im Z^{N}} is G_{ℝ}-invariant and strictly plurisubharmonic. Moreover, ϕ is a weak exhaustion of H^{N}.

The simplest way to see that ϕ is strictly plurisubharmonic is to note that Z^{j} → \frac{1}{det Im Z^{j}} it is given by the Bergmann kernel function on H. Since det Im Z = 0 for Z ∈ ∂H, ϕ is a weak exhaustion of H^{N}, i.e., ϕ(Z_{k}) → +∞ if lim Z_{k} = Z_{0} ∈ ∂(H^{N}) ⊂ V^{N}.

Let K_{ℝ} := {(a, ā); a ∈ SU_{2}(ℂ)} be the maximal compact subgroup of G_{ℝ}. We set V^{0} := {Z ∈ V; det Z ≠ 0}. Note that V//G ≅ ℂ and that after this identification the quotient map is given by det : V → ℂ. In particular, V^{0} is saturated with respect to V → V//G.

Lemma 1. Let (W_{n}) be a sequence in H such that (π(W_{n})) converges in V//G. Then there exist h_{n} ∈ G_{ℝ} such that a subsequence of (h_{n} ∗ W_{n}) converges in V.

Proof. There exist u_{n} ∈ K_{ℝ} such that

X_{n} := u_{n} ∗ W_{n} =: ({x_{n} \atop 0} {z_{n} \atop y_{n}} ) .

Since (π(W_{n})) = (π(X_{n})) converges, it follows that | det X_{n}| = |x_{n}y_{n}| ≤ R for some R ≥ 0 and all n. Furthermore, X_{n} ∈ H implies that \frac{1}{4}|z_{n}|^{2} < Im x_{n}Im y_{n} ≤ |x_{n}y_{n}| = | det X_{n}|. Therefore (z_{n}) is bounded. Now 0 < |x_{n}y_{n}| ≤ R implies that |r_{n}^{2}x_{n}| = |r_{n}^{−2}y_{n}| for some r_{n} > 0. In particular the sequence (r_{n}^{2}x_{n}, r_{n}^{−2}y_{n}) is bounded. Hence h_{n} ∗ W_{n} has a convergent subsequence where h_{n} := r_{n} · u_{n} ∈ G_{ℝ} and r_{n} is identified

with (({r_{n} \atop 0} 0_{\frac{1}{rn}} ), ({r_{n} \atop 0} 0_{\frac{1}{rn}} )). □

Remark 1. Geometrically Lemma 1 asserts that H is relatively compact over V//G mod G_{ℝ}.

Lemma 2. Let (Z_{n}, W_{n}) be a sequence of points in H × H and assume that

(i) π(Z_{n}, W_{n}) converges in (V × V)//G and (ii) W_{0} = lim W_{n} exists in H. Then a subsequence of (Z_{n}) converges to a Z_{0} ∈ \overline{H}.

Proof. Note that V × V^{0} is an open G-stable subset of V × V which is saturated with respect to V × V → (V × V)//G and contains H × H. The map V × V^{0} → V, (Z, W) → ZW^{−1}, is G-equivariant, where G acts on the image V by conjugation with the first component, i.e. by int(g, h) · X = gXg^{−1}. It is sufficient to show the following

Claim. A subsequence of (X_{n}) converges.

Since the image of X_{n} := Z_{n}W_{n}^{−1} in V//int G converges, the trace and the determinant of X_{n} and therefore the eigenvalues of X_{n} are bounded. Let u_{n} = (a_{n}, ā_{n}) ∈ K_{ℝ} be such that int a_{n} · X_{n} = (u_{n} ∗ Z_{n})(u_{n} ∗ W_{n})^{−1} = ({x_{n} \atop 0} {z_{n} \atop y_{n}} ). Since K_{ℝ} is compact, we may assume that X_{n} = ({x_{n} \atop 0} {z_{n} \atop y_{n}} ).

Let W_{n} =: ({a_{n} \atop c_{n}} {b_{n} \atop d_{n}} ) and W_{0} =: ({a_{0} \atop c_{0}} {b_{0} \atop d_{0}} ). By assumption we have W_{0} ∈ H. Therefore Im d_{0} ≠ 0. From

Z_{n} = X_{n}W_{n} = ( \begin{matrix}x_{n}a_{n} + z_{n}c_{n} & x_{n}b_{n} + z_{n}d_{n} \\ y_{n}c_{n} & y_{n}d_{n} \end{matrix} ) ∈ H

it follows that

\frac{1}{|z_{n}|^{2}} (Im (x_{n}a_{n} + z_{n}c_{n})Im (y_{n}d_{n}) − \frac{1}{4}|x_{n}b_{n} + z_{n}d_{n} − ȳ_{n}c̄_{n}|^{2}) > 0

for z_{n} ≠ 0. Since the eigenvalues x_{n}, y_{n} and a_{n}, b_{n}, c_{n}, d_{n} are bounded, d_{0} ≠ 0 implies that |z_{n}| is bounded. Thus (X_{n}) has a convergent subsequence. □

Remark 2. The proofs of Lemma 1 and Lemma 2 use arguments which can be found at least implicitly in [Z] on p. 17.

In the above proof we used that H ⊂ V^{0} which is implied by det Im Z ≤ | det Z|.

Corollary 1. If Z_{n} = (Z_{n}^{1}, …, Z_{n}^{N}) ∈ H^{N} are such that (π(Z_{n})) converges in V^{N}//G and (Z_{n}^{N}) converges in H, then (Z_{n}) has a convergent subsequence in \overline{H}^{N}. □

Lemma 3. ϕ is a weak exhaustion of X over V//G.

Proof. Let (Z_{n}) = ((Z_{n}^{1}, …, Z_{n}^{N})) be a sequence in H^{N} such that q := lim π(Z_{n}) ∈ ∂(G∗H^{N}//G) ⊂ V^{N}//G exists. There are h_{n} ∈ G_{ℝ} such that a subsequence of (h∗Z_{n}^{N}) converges to W^{N} ∈ \overline{H} (Lemma 1). Now, if W^{N} ∈ ∂H, then lim sup ϕ(Z_{n}) = +∞. Thus assume that W^{N} ∈ H. It follows that (h_{n} ∗ Z_{n}) has a subsequence which converges to W ∈ \overline{H}^{N} (Corollary 1). But W is not in H^{N}, since q = π(W) ∈ ∂(G ∗ H^{N}//G). Thus W ∈ ∂H^{N} and therefore again lim sup ϕ(Z_{n}) = +∞ follows. □

Lemma 4. The function ϕ is an exhaustion mod G_{ℝ} along π.

Proof. For r > 0 let Z_{n} ∈ H^{N}, Z_{n} =: (Z_{n}^{1}, …, Z_{n}^{N}), be such that ϕ(Z_{n}) ≤ r and assume that lim π(Z_{n}) exists in G ∗ H^{N}//G. Thus there are h_{n} ∈ G_{ℝ} such that (h_{n} ∗ Z_{n}^{N}) has a subsequence which converges to some W^{N} ∈ \overline{H}. If W^{N} ∈ ∂H, then ϕ(Z_{n}) goes to infinity. This contradicts ϕ(h_{n} ∗ Z_{n}) ≤ r. Thus W^{N} ∈ H and therefore (h_{n} ∗ Z_{n}) has a subsequence with limit W = (W^{1}, …, W^{N}) ∈ \overline{H}^{N}. The same argument as above implies that W^{j} ∈ H for j = 1, …, N. □

Proof of the Theorem. From the invariant theoretical point of view the G = SL_{2}(ℂ) × SL_{2}(ℂ) action on V^{N} is the N-fold product of the standard representation of SO_{4}(ℂ) on ℂ^{4}. It is well known that for any N = 1, 2, … the generic G-orbit in V^{N} is closed. Let (V^{N})^{0} denote the set of points in V^{N} which lie in a generic closed orbit, i.e., (V^{N})^{0} is a union of the fibres of the quotient V → V//G which consist exactly of one G-orbit. Since the G_{ℝ}-action on H is proper, G_{ℝ} acts properly on H^{N}. It follows from the results in §2 that there is a G-invariant plurisubharmonic function Ψ on G ∗ H^{N} which is a weak exhaustion. Thus G ∗ H^{N} is a domain of holomorphy. □

Corollary 2. The image G ∗ H^{N}//G of H^{N} in V^{N}//G is an open Stein subspace. □

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