McGill University, December 4 - 7, 2015
This is joint work with Nan Li.
I will explain this result and some of its implications to Riemannian geometry of four-dimensional manifolds. For example, for each closed four-dimensional Riemannian manifold $M$ and each sufficiently small positive $\epsilon$ the set of isometry classes of Riemannian metrics on $M$ of volume one with the injectivity radius greater than $\epsilon$ is disconnected. A similar disconnectedness result holds for sets of Riemannian structures with $\sup|K|diam^2 \le x$ on each closed four-dimensional manifold with non-zero Euler characteristic providing that $x$ is sufficiently large. (A joint work with Boris Lishak.)
For the most part this existence question has been a primary focus of research. However, there is an equally intruiging secondary question. If a manifold admits a given type of metric, how are such metrics distributed among all possible Riemannian metrics on this object? For example are they rare or common? How 'many' metrics and geometries does a given manifold allow for?
To answer these questions, one usually looks at the space of metrics satisfying various given curvature conditions on the manifold, or its quotient by the group of diffeomorphisms, the so-called moduli space of metrics, and studies its respective properties.
In my talk, I will survey and describe recent progress
on these questions, focusing primarily on connectedness properties
of moduli spaces of nonnegative sectional curvature metrics.