Moment maps for torus actions
This paper examines when a moment map exists for a symplectic torus action on a compact manifold. Having in mind that a moment map for a \(G\)-action restricts to a moment map for the action of a Lie subgroup \(H\subset G\), then, nonexistence results for \(S^1\)-actions extend to nonexistence results for torus actions. The author shows that a fixed-point-free action of \(S^1\) on a compact, connected manifold cannot have a moment map. As an application of this together with the Poincaré-Hopf index theorem, a partial converse of a result of Marsden and Weinstein can be shown. Marsden and Weinstein showed that if \(H^1(M, Q)= 0\) then a torus action on \(M\) has a moment map. The partial converse shown in the paper states that if \(S^1\) acts symplectically on a compact surface \(M\) and \(H^1(M, Q)\neq 0\) then there can be no moment map for the action. A surface of revolution, \(M\subset\mathbb{R}^3\), considered as a symplectic manifold with symplectic form the area form, has a natural \(S^1\) action with orbits the parallels of the surface. After discussing the incorrectness of an obvious choice for a moment map, the author gives a true moment map that admits a geometric interpretation: If the surface of revolution is given in cylindrical coordinates by \(r= f(z)\), then the function \(\mu(r_0, \theta_0, z_0)\) given by the area of the part of the surface with \(z< z_0\) is a moment map.
scientific article; zbMATH DE number 894689