Let a torus act effectively on a compact connected cooriented contact manifold, and let be the natural momentum map on the symplectization. We prove that, if is bigger than 2, the union of the origin with the image of is a convex polyhedral cone, the nonzero level sets of 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 .
River Chiang. Yael Karshon. "Convexity package for momentum maps on contact manifolds." Algebr. Geom. Topol. 10 (2) 925 - 977, 2010. https://doi.org/10.2140/agt.2010.10.925