Rob Kusner Department of Mathematics, University of Massachusetts, Amherst, MA 01003 and School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, NJ, 08540 Natural Lagrangian and Symplectic Structures for Constant Mean Curvature Moduli Spaces and Related Geometric Variational Problems

We investigate a natural symplectic manifold — a configuration space of weighted, parametrized geodesics — into which the moduli space of certain properly embedded constant mean curvature (CMC) hypersurfaces in Euclidean or hyperbolic space (or, of complete, conformally flat, positive constant scalar curvature (CSC) metrics on the complement of certain sets in the n-sphere) maps with isotropic image. This map α, which assigns to each CMC hypersurface (or CSC metric) the geodesic axes, weights and phases of the (asymptotically Delaunay) ends, is a kind of ‘Gauss map’ for the moduli spaces. We also introduce a natural Lagrangian submanifold — the space of ends of balanced geodesic trees — that serves as model for the image of the CMC (or CSC) moduli spaces under certain nondegeneracy hypotheses. This approach allows us to separate the analytic issues (such as those concerning the nondegeneracy hypotheses) from questions about the geometry of the moduli spaces themselves. By focusing on the latter, particularly the geometry of the target and image of α, we provide a common framework for understanding both the CMC and CSC cases, and thus clarify further some of the curious analogies between these cases.