Minimal Surface Entropy for Negatively Curved N-manifolds and Rigidity

We focus on an odd-dimensional closed manifold M that admits a hyperbolic metric. For any metric on M with sectional curvature less than or equal to -1, we introduce the minimal surface entropy to count the number of surface subgroups. It attains the minimum if and only if the metric is hyperbolic. This result is an extension of the work on 3-manifolds by Calegari-Marques-Neves. I will introduce their ideas and discuss the problems and solutions for higher dimensions. If time permits, I will mention the results for locally symmetric spaces and hyperbolic 3-manifolds of finite volume.