L2-cohomology and Complete Hamiltonian Manifolds

A classical theorem of Frankel for compact Kähler manifolds states that a Kähler S-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when Hodge theory holds on non-compact manifolds, then Frankel’s theorem still holds. Finally, we present several concrete situations in which the assumptions of the metatheorem hold. 1. The Classical Frankel Theorem An S1-action on a symplectic manifold (M,ω) is Hamiltonian if there exists a smooth map, the momentum map, μ : M → (s1)∗ ' R into the dual (s1)∗ of the Lie algebra s1 ∼= R of S1, such that iξMω := ω(ξM , ·) = dμ, for some generator ξ of s1, that is, the 1-form iξMω is exact. Here ξM is the vector field on M whose flow is given by R ×M 3 (t,m) 7→ eitξ ·m ∈ M , where the dot denotes the S1-action on M . If (M,ω) is connected, compact and Kähler, the following result of T. Frankel is well-known: Frankel’s Theorem ([Fr59]). Let M be a compact connected Kähler manifold admitting an S1-action preserving the Kähler structure. If the S1-action has fixed points, then the action is Hamiltonian. This theorem generalizes in various ways; for example, the S1-action may be replaced by a G-action, where G is any compact Lie group and the Kähler structure may be weakened to a symplectic structure. The purpose of this paper is to generalize Frankel’s theorem to certain noncompact complete Riemannian manifolds. More specifically, we describe a set of hypotheses 1 Partially supported by NSF grant DMS-1105050 2 Partially supported by NSF Grant DMS-0635607, NSF CAREER Award 1055897, Spanish Ministry Grant MTM 2010-21186-C02-01, and the MSRI 2010-11 program Symplectic and Contact Geometry and Topology. Partially supported by the government grant of the Russian Federation for support of research projects implemented by leading scientists, Lomonosov Moscow State University, under agreement No. 11.G34.31.0054, by Swiss NSF grant 200021-140238, and the 2010-11 MSRI program Symplectic and Contact Geometry and Topology.