Katok's entropy conjecture near real and complex hyperbolic metrics

We show that, given a real or complex hyperbolic metric $g_0$ on a closed manifold $M$ of dimension $n\geq 3$, there exists a neighborhood $\mathcal U$ of $g_0$ in the space of negatively curved metrics such that for any $g\in \mathcal U$, the topological entropy and Liouville entropy of $g$ coincide if and only if $g$ and $g_0$ are homothetic. This provides a partial answer to Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we obtain a local rigidity result for the hyperbolic rank and for metrics with $C^2$ Anosov foliations near complex hyperbolic metrics.