Openness and convexity for momentum maps

Birtea, Petre; Ortega, Juan-Pablo; Ratiu, Tudor

doi:10.1090/S0002-9947-08-04689-8

Openness and convexity for momentum maps
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by Petre Birtea, Juan-Pablo Ortega and Tudor S. Ratiu PDF

Trans. Amer. Math. Soc. 361 (2009), 603-630 Request permission

Abstract:

The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of a topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so-called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the so-called Local-to-Global Principle that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to noncompact manifolds.

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