Average Area Ratio and Minimal Surface Entropy of Hyperbolic Manifolds

This thesis studies the average area ratio and minimal surface entropy of hyperbolic manifolds. On closed hyperbolic manifolds of dimension $n\geq 3$, we review thedefinition of the average area ratio of a metric $h$ whose scalar curvature is bounded below by $-n(n-1)$ in comparison to the hyperbolic metric $h_0$. We prove that it reaches its local minimum value of one at $h_0$, which solves a localized version of Gromov's conjecture. Furthermore, in the case of odd $n$, assuming $h$ is a metric with sectional curvature no greater than $-1$, we introduce the concept of minimal surface entropy of $h$, which quantifies the number of surface subgroups. It achieves its minimum value if and only if the metric is hyperbolic. Additionally, we explore the relationship between the average area ratio and the normalized total scalar curvature for hyperbolic $n$-manifolds. We also discuss its connection to the minimal surface entropy when $n$ is odd.