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Moment maps and diffeomorphisms. (English) Zbl 1106.53037

Yau, S.T. (ed.), Surveys in differential geometry. Papers dedicated to Atiyah, Bott, Hirzebruch and Singer. Somerville, MA: International Press (ISBN 1-57146-069-1/hbk). Surv. Differ. Geom., Suppl. J. Differ. Geom. 7, 107-127 (2000).
The paper starts with recalling the crucial observation by Atiyah and Bott that the curvature of a connection on a bundle over a surface can be viewed as the “momentum” corresponding to the action of the gauge group. The purpose of the present paper, as it is formulated in the text, is “to explore some similar ideas in the framework of diffeomorphism groups”.
Given a compact manifold \(S\) with a fixed volume form and a symplectic manifold \(M\), the space \({\mathcal M}\) of smooth maps from \(S\) to \(M\) in some homotopy class is considered as an infinite-dimensional symplectic manifold. The Lie group \({\mathcal G}\) of volume preserving diffeomorphisms of \(G\) acts symplectically on this infinite-dimensional manifold. The author constructs an equivariant moment map for this action under assumption that \(H^1(S)=0\) and \(f^\ast[\omega] = 0\) in \(H^2(S)\) where \(\omega\) is the symplectic form on \(M\) and shows how this construction leads to a non-equivariant moment map when these assumptions fall. If \(S\) is a symplectic manifold also then another construction leads an equivariant moment map for the group of symplectomorphisms of \(S\) and if \(H^1(S)\) it still gives such a map for the group of exact symplectomorphisms of \(S\).
By using these constructions the moment map theory is applied to a description of some moduli spaces and, in particular, of a torus \(T = H_1(S,{\mathbb R})/H_1(S,{\mathbb Z})\) bundle over the moduli space of special Lagrangian submanifolds of a Calabi–Yau manifold. This space is realized as a Kähler quotient and this construction is generalized for torus bundles over the moduli spaces of LS-submanifolds in complex symplectic manifolds.
In the other part of the paper the author considers the gradient flow for the norm of the moment map. Assuming that \(S\) and \(M\) are diffeomorphic Riemann surfaces and the infinite-dimensional manifold \({\mathcal M}\) is formed by oriented diffeomorphisms it is showed that solutions of such a flow exist for all time and converge to area-preserving diffeomorphisms. For \(S\) a Riemann surface and \(M\) a complex plane with the standard symplectic structure such a flow is the reverse porous-medium equation. If \(S\) is a Riemann surface and \(M\) is hyper-Kähler manifold then by using such a flow it is proved that an immersion \(f: S \to M\) is a critical point of the energy functional iff its image is a minimal surface and the Riemannian area is a constant multiple of the volume form on \(S\). Some other applications and also open problems related to these flows and their applications are presented.
For the entire collection see [Zbl 1044.53002].

MSC:

53C38 Calibrations and calibrated geometries
53D20 Momentum maps; symplectic reduction
53D12 Lagrangian submanifolds; Maslov index
37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C28 Twistor methods in differential geometry
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