Counting essential minimal surfaces in closed negatively curved n-manifolds

TL;DR

This paper introduces a new entropy measure for counting minimal surface subgroups in negatively curved manifolds, showing it uniquely identifies hyperbolic metrics and computing it for various symmetric spaces.

Contribution

It defines the minimal surface entropy for negatively curved manifolds and characterizes hyperbolic metrics as the unique minimizers, also computing entropy for other symmetric spaces.

Findings

Minimal surface entropy attains minimum only for hyperbolic metrics.

Explicit entropy values computed for locally symmetric spaces.

Provides a new invariant for classifying negatively curved manifolds.

Abstract

For closed odd-dimensional manifolds with sectional curvature less or equal than -1, we define the minimal surface entropy that counts the number of surface subgroups. It attains the minimum if and only if the metric is hyperbolic. Moreover, we compute the entropy associated with other locally symmetric spaces and cusped hyperbolic 3-manifolds.