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The minimum principle from a Hamiltonian point of view. (English) Zbl 0939.32021
Suppose $$G_{\mathbb R}$$ is a connected real form of a complex Lie group $$G$$. Let $$Z$$ be a complex $$G$$-manifold containing a $$G_{\mathbb R}$$-stable domain $$X$$ such that $$G\cdot X = Z$$. The basic question considered in this paper is to find conditions under which $$Z$$ is a domain of holomorphy if $$X$$ is a domain of holomorphy. In the setting where $$Z$$ is a domain in a Stein space $$V$$ one can construct $$G$$-invariant plurisubharmonic functions starting with $$G_{\mathbb R}$$-invariant plurisubharmonic functions on $$X$$. (The goal of proving $$Z$$ Stein is achieved by finding a plurisubharmonic function on $$Z$$ which goes to $$+\infty$$ as one approaches every boundary point of $$Z$$ in $$V$$.) This is accomplished under the assumption that there exists a geometric $$G$$-quotient $$\pi : Z \to Z/G$$ by taking a smooth $$G_{\mathbb R}$$-invariant plurisubharmonic function $$\varphi$$ on $$X$$ which is a strong plurisubharmonic exhaustion on each fiber of $$\pi|_{X}$$ and defining a function on $$Z/G$$ which is the minimum of $$\varphi$$ on each fiber, an idea due to J.-J. Loeb [Ann. Inst. Fourier 35, No. 4, 59-97 (1985; Zbl 0563.32013)] in the case of free actions. If $$\varphi:X \to {\mathbb R}$$ is a smooth $$G_{\mathbb R}$$-invariant strictly plurisubharmonic function which is an exhaustion mod $$G_{\mathbb R}$$ along $$\pi$$, then there is an associated invariant moment map $$\mu : X \to{\mathfrak g}_{\mathbb R}^{*}$$. The set $$\mu^{-1}(0)$$ is the set of fiberwise critical points of $$\varphi$$. This compatibility of the symplectic reduction, i.e., forming $$\mu^{-1}(0)/G_{\mathbb R}$$, and the minimum principle, shown to hold when $$G_{\mathbb R}$$ is compact by P. Heinzner, A. T. Huckleberry and F. Loose [J. Reine Angew. Math. 455, 123-140 (1994; Zbl 0803.53042)] is proved here under certain precise conditions. This is then used to make the ideas outlined above work.
As an application the author shows that the extended future tube is a domain of holomorphy; this problem has also been considered by X.-Y. Zhou [“On the extended future tube conjecture”, preprint; per bibl.].

##### MSC:
 32M05 Complex Lie groups, group actions on complex spaces
##### Keywords:
symplectic reduction; minimum principle
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