The minimum principle from a Hamiltonian point of view

Abstract

Let $G$ be a complex Lie group and $G_{\mathbb{R}}$ a real form of $G$. For a $G_{\mathbb{R}}$-stable domain of holomorphy $X$ in a complex $G$-manifold we consider the question under which conditions the extended domain $G\cdot X$ is a domain of holomorphy. We give an answer in term of $G_{\mathbb{R}}$-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 $G\cdot X$ is a domain of holomorphy in the case where $X$ is the $N$-fold product of the tube domain in ${\mathbb{C}}^4$ over the positive light cone and $G$ is the connected complex Lorentz group acting diagonally.