Moduli Spaces of Metrics in Geometry and Physics

Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature or other geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, of special holonomy, etc.

A natural question to ponder, which is also of relevance in parts of physics like general relativity, string and mirror symmetry, is then what the space of all such metrics does look like. Moreover, one can also study this question for corresponding moduli spaces of metrics, i.e., for quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics. These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties hence provide the right means to measure 'how many' different metrics and ' how many' geometries the given manifold actually does exhibit.

In my talk, I will report on some general results and open questions about the global structure and topology of such spaces and moduli spaces, with a focus on Ricci flat and non-negative Ricci or sectional curvature metrics, since these play important roles in both Riemannian and complex geometry as well as in the investigation of Calabi-Yau manifolds and their metric degenerations under collapse. Moreover, if time permits, I will also discuss broader non-traditional approaches from metric geometry and analysis to these objects under the presence of lower curvature bounds, and the Lorentzian instead of Riemannian setting.