Appears in collection : Not Only Scalar Curvature Seminar

There is by now a broad body of work on the topology of Riemannian manifolds with positive scalar curvature, and it remains a flourishing field of study to this day. Much less is known about negative scalar curvature lower bounds. Here one seeks to pass from a negative scalar curvature lower bound, given knowledge of the ambient topology, to a statement about some “geometric feature” of the ambient Riemannian manifold. After giving a brief overview of what is known, I will focus in this talk on the case where the “geometric feature” in question is a quantity measuring the asymptotic growth rate of minimal surfaces, counted by area. The latter was introduced by Calegari-Marques-Neves, and is motivated by the theory of the geodesic flow of a compact negatively curved Riemannian manifold, one of the hallmark examples of a chaotic dynamical system. In contrast to the case of geodesics, however, tools from geometric analysis such as the Ricci flow are important when one seeks to obtain analogous results for minimal surfaces. I will give a gentle overview of the key ideas here while avoiding technical details. Finally, I will discuss an application to the following general question, suitably formulated: to what extent is a Riemannian manifold determined by the minimal surfaces that it contains? Based on joint work with Andre Neves, and Yangyang Li and James Marshall Reber.