# Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

## TL;DR

This paper investigates minimal surface entropy in finite-volume hyperbolic 3-manifolds, establishing conditions under which the hyperbolic metric minimizes or maximizes entropy, and analyzing Ricci flow convergence to hyperbolic metrics.

## Contribution

It provides new characterizations of hyperbolic metrics as entropy minimizers or maximizers under various curvature bounds and convergence conditions, linking Ricci flow dynamics to entropy properties.

## Key findings

- Entropy minimized only by hyperbolic metrics under certain curvature bounds.
- Hyperbolic metric maximizes entropy among metrics with scalar curvature ≥ -6.
- Ricci flow convergence rate relates to entropy optimization in hyperbolic 3-manifolds.

## Abstract

This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic $3$-manifolds of finite volume, particularly its comparison to the hyperbolic minimal surface entropy in terms of sectional and scalar curvature. On one hand, for metrics that are bilipschitz equivalent to the hyperbolic metric and have sectional curvature bounded above by $-1$ and uniformly bounded below, we show that the entropy achieves its minimum if and only if the metric is hyperbolic. On the other hand, by analyzing the convergence rate of the Ricci flow toward the hyperbolic metric, we prove that among all metrics with scalar curvature bounded below by $-6$ and with non-positive sectional curvature on the cusps, the entropy is maximized at the hyperbolic metric, provided that it is infinitesimally rigid. Furthermore, if the metrics are uniformly $C^0$-close to the hyperbolic metric and asymptotically cusped, then the entropy associated with the Lebesgue measure is uniquely maximized at the hyperbolic metric.