Convexity package for momentum maps on contact manifolds

TL;DR

This paper proves convexity and connectedness properties of momentum maps on contact manifolds with torus actions, extending previous results and analyzing low-dimensional cases.

Contribution

It establishes convexity and connectedness results for momentum maps on contact manifolds with torus actions when the dimension exceeds two, answering a question by Eugene Lerman.

Findings

Union of the origin with the image of the momentum map forms a convex polyhedral cone.

Non-zero level sets of the momentum map are connected.

The momentum map is open onto its image.

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 dim T > 2, the union of the origin with the image of Psi is a convex polyhedral cone, the non-zero 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 dim T <= 2.