Convexity properties of the moment map for certain non-compact manifolds
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.
scientific article; zbMATH DE number 820219