Convexity properties of the moment map for certain non-compact manifolds
type: Item
instance of: scholarly article
title: Convexity properties of the moment map for certain non-compact manifolds
zbMATH Open document ID: 0842.58034
author: Elisa Prato
published in: Communications in Analysis and Geometry
publication date: 1996-01-01T00:00:00Z
review text: The author considers a connected symplectic manifold \(M\) and a Hamiltonian action of a torus \(T\) with corresponding moment map \(\Phi : M \to {\mathfrak t}^*\). For compact \(M\), Atiyah and Guillemin and Sternberg have proved that \(\Phi(M)\) is the convex hull of the image of the fixed points of the torus action. Here, in the non-compact case, the author makes the supplementary assumption that \(\mathfrak t\) contains an integral element \(X_0\) such that \(\Phi_{X_0}\) is a proper function with a unique critical value which is a minimum. The main result states that under this assumption \(\Phi(M)\) is the convex hull of a finite number of affine rays. As a nice application, a proof is obtained of a convexity theory by \textit{S. M. Paneitz} [J. Funct. Anal. 43, 313-359 (1981; Zbl 0476.22009)] concerning positive Kähler elliptic co-adjoint orbits.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 820219
zbMATH Keywords: symplectic manifold
zbMATH Keywords: Hamiltonian action
zbMATH Keywords: moment map
zbMATH Keywords: convexity
reviewed by: W. Sarlet
MaRDI profile type: Publication
DOI: 10.4310/CAG.1994.V2.N2.A5
description: scientific article; zbMATH DE number 820219