Convexity package for momentum maps on contact manifolds

Let a torus $T$ act effectively on a compact connected cooriented contact manifold, and let $Ψ$ be the natural momentum map on the symplectization. We prove that, if $dim T$ is bigger than 2, the union of the origin with the image of $Ψ$ is a convex polyhedral cone, the nonzero level sets of $Ψ$ 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 $dim T \leq 2$ .

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Keywords

momentum map, contact manifold, torus action, convexity

Mathematical Subject Classification 2000

Primary: 53D10, 53D20

Secondary: 52B99

References

Publication

Received: 5 November 2009

Revised: 25 February 2010

Accepted: 2 March 2010

Published: 17 April 2010

Authors

River Chiang
Department of Mathematics National Cheng Kung University Tainan 701 Taiwan

Yael Karshon
Department of Mathematics University of Toronto Toronto, Ontario M5S 2E4 Canada
http://www.math.toronto.edu/karshon/

Volume 10, issue 2 (2010)

River Chiang and Yael Karshon

Abstract