# Entropy Rigidity of negatively curved manifolds of finite volume

## TL;DR

This paper establishes that negatively curved finite volume manifolds with maximal entropy are hyperbolic, and shows that sharing the same length spectrum implies isometry to a hyperbolic manifold, extending classical compact case results.

## Contribution

It proves an entropy-rigidity theorem for finite volume negatively curved manifolds and provides a new proof for the spectral rigidity result without using entropy.

## Key findings

- Maximal entropy characterizes hyperbolic manifolds.
- Finite volume manifolds with same length spectrum are isometric to hyperbolic ones.
- Extension of classical compact case theorems to finite volume setting.

## Abstract

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we also give a direct, entropy-free proof of this fact). We compare with the classical theorems holding in the compact case, pointing out the main difficulties to extend them to finite volume manifolds.