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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.].

32M05 Complex Lie groups, group actions on complex spaces
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