The minimum principle from a Hamiltonian point of view

TL;DR

This paper proves that under certain conditions, the extended domain G.X remains a domain of holomorphy, and applies this to confirm the extended future tube conjecture involving complex Lorentz groups and tube domains.

Contribution

It introduces a Hamiltonian perspective to the minimum principle, providing a new proof of the extended future tube conjecture in complex Lie group actions.

Findings

Extended domain G.X is a domain of holomorphy under specified conditions.

Proof of the extended future tube conjecture for complex Lorentz groups.

Establishment of a Hamiltonian approach to the minimum principle.

Abstract

Let G be a complex Lie group, G_R a real form of G and X a G_R-stable domain of holomorphy in a complex G-manifold. If there is a G_R-invariant strictly plurisubharmonic function on X which has certain exhaustion properties, then we show that the extended domain G.X is also a domain of holomorphy. As an application we give a proof of the extended future tube conjecture. This is the assertion 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 in R^4 and G is the connected complex Lorentz group acting diagonally.