Flexibility of geometrical and dynamical data in fixed conformal classes

Abstract: Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $\sigma$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area $A$ conformally equivalent to $\sigma$. For such metrics, we show that the diameter is bounded above and the Laplace spectrum is bounded below away from zero by constants which depend on $\sigma$. On the other hand, we prove that the metric entropy of the geodesic flow with respect to the Liouville measure is flexible. Consequently, we also provide the first known example showing that the bottom of the $L^2$-spectrum of the Laplacian cannot be bounded from above by a function of the metric entropy. We also provide examples showing that our conditions are essential for the established bounds.