What manifolds support metrics of positive scalar curvature? If so, what can one say about the associated moduli space? These are two fundamental problems in Riemannian Geometry, for which great progress has been made over the last fifty years, but that are still highly elusive and far from being fully resolved.
Partly motivated by the study of initial data sets for the Einstein equations in General Relativity, I will present some results that aim at moving one step further, studying the interplay between two different curvature conditions, given by pointwise inequalities on the scalar curvature of a manifold, and the mean curvature of its boundary.
In particular, after a broad contextualization, I will focus on recent joint work with Chao Li (Princeton University), where we give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any such case, we prove that the associated moduli space of metrics is path-connected. We can also refine our methods so to construct continuous paths of non-negative scalar curvature metrics with minimal boundary, and to obtain analogous conclusions in that context as well. In particular, note that we can derive the path-connectedness of asymptotically flat scalar flat Riemannian 3-manifolds with minimal boundary, which goes in the direction of understanding the space of vacuum black-hole solutions to the Einstein field equations.
Our work relies on a combination of earlier fundamental contributions by Gromov-Lawson and Schoen-Yau, on the smoothing procedure designed by Miao to handle singular interfaces, and on the interplay of Perelman's Ricci flow with surgery and conformal deformation techniques introduced by Codá Marques in dealing with the closed case.