Flexibility of entropies for surfaces of negative curvature

Abstract: We consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a Riemannian metric $g$ of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for $g$ is greater than or equal to the topological entropy for the metric of constant negative curvature on $M$ with the same total area which is greater than or equal to the metric entropy with respect to the Liouville measure of geodesic flow for $g$. Equality holds only in the case of constant negative curvature. We prove that those are the only restrictions on the values of topological and metric entropies for metrics of negative curvature.