Convexity package for momentum maps on contact manifolds

Abstract

Let a torus $T$ act effectively on a compact connected cooriented contact manifold, and let $\Psi$ be the natural momentum map on the symplectization. We prove that, if $dimT$ is bigger than 2, the union of the origin with the image of $\Psi$ is a convex polyhedral cone, the nonzero level sets of $\Psi$ are connected (while the zero level set can be disconnected), and the momentum map is open as a map to its image. This answers a question posed by Eugene Lerman, who proved similar results when the zero level set is empty. We also analyze examples with $dimT\le 2$.