Submanifold averaging in Riemannian and symplectic geometry (Q2496877)

In 1999 \textit{A. Weinstein} [J. Eur. Math. Soc. (JEMS) 2, No. 1, 53--86 (2000; Zbl 0957.53021)] presented a procedure to average a family \(N_g\) of submanifolds of a Riemannian manifold \(M\). In the first part of the present paper a result by H. Karcher and the author is exhibited which improves Weinstein's theorem: the compactness assumption on the \(N_g\)'s is superfluous. In the main body of the paper Weinstein's averaging is specialized to the setting of symplectic geometry. The main theorem is as follows. Let \((M^m,g,\omega,I)\) be an almost-Kähler manifold satisfying \(|\nabla\omega|< 1\) and \(\{N^n_g\}\) a family of isotropic submanifolds of \(M\) parametrized in a measurable way by elements of a probability space \(G\), such that all the pairs \((M, N_g)\) are gentle. If \(d_1(N_g,N_h)< \varepsilon< 1/70000\) for all \(g\) and \(h\) in \(G\), there is a well defined isotropic center of mass submanifold \(L^n\) with \(d_0(N_g,L)< 1000\varepsilon\) for all \(g\) in \(G\). This construction is equivariant with respect to isometric symplectomorphisms of \(M\) and measure preserving automorphisms of \(G\). One consequence is a statement about group actions. As a simple application this statement is applied to almost invariant isotropic submanifolds of a Hamiltonian \(G\)-space; some information is deduced about their images under the moment map.