Entropy Rigidity of negatively curved manifolds of finite volume

Abstract

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\le K_X\le -1$, then $En{t}_{top}(X)=n-1$ 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.

20 p. arXiv admin note: text overlap with arXiv:1503.03971