Curvature constraints and space of metrics – CCEM

A fundamental problem in Riemannian geometry is to understand `spaces of metrics’ defined by putting constraints on certain geometric quantities, in particular various notions of curvature. One studies either a set of Riemannian metrics on a fixed manifold, or a class of Riemannian manifolds. Those `spaces’ can be endowed with topologies, such as the Gromov-Hausdorff topology. When the space is not compact, it is natural to try to introduce singular metrics which can occur as limits, a simple example being a cone in Euclidean space, which is a limit of smooth surfaces whose Gauss curvature fails to be bounded from above. This motivates the introduction of several classes of singular metric spaces, some old, such as Alexandrov spaces in the 1950s (geodesic spaces of curvature bounded from below in the synthetic sense of having fat triangles), Cheeger-Colding spaces, and more recently the spaces introduced by Lott-Villani/Sturm and Gigli-Ambrosio-Savaré. Those spaces can be studied in connection with smooth Riemannian manifolds, but also for their own sake.
In this project, we shall work on several types of questions : topological constraints implied by hypotheses on the curvature, existence and/or uniqueness of a « best metric » in a given class, homotopy type of a space of metrics. These questions are strongly connected to the topic of geometric flows, in particular Ricci flow and Mean Curvature Flow. On the one hand, the notion of Gromov-Hausdorff convergence allows to give meaning to Ricci flow with nonsmooth initial condition, which permits in some cases to smoothen singular spaces. On the other hand, the study of singularities of these flows uses convergence arguments via compactness theorems and the blow-up technique. Controlling the singularities is at the core of the construction of surgically modified solutions, the most well-known example being Perelman’s use of Ricci flow to prove the Geometrization Conjecture. It also enabled Marques and Bessières-Besson-Maillot-Marques to prove connectedness result of spaces of metrics with positive scalar curvature. We hope to develop similar techniques for surfaces embedded in 3-manifolds using Mean Curvature Flow.
An important part of the project consists in studying a special class of singular metric spaces : stratified spaces endowed with metrics of iterated conical type. This is a fairly large class on which ideas can be tested so as to better understand the more general spaces mentioned above. For instance, we hope to prove that the stratified metric with constant curvature on the celebrated Gromov-Thurston manifolds is the unique Einstein metric. It was previously thought that those could admit other (smooth) Einstein metrics. This would shed some light on the question posed by Besse 30 years ago in the book Einstein manifolds : « Are manifolds admitting an Einstein metric rather scarce or numerous? » 
In another part of the project we hope to extend the theory of spaces with Ricci curvature bounds, due to Cheeger, Colding, Naber and their co-authors, under weaker hypotheses involving scalar curvature or integral curvature bounds. We shall also endeavor to work with coarser quantities, such as entropy, instead of curvature. Besson-Courtois-Gallot-Sambusetti have already obtained striking results with this approach.