  We show the flexibility of the metric entropy and obtain additional
restrictions on the topological entropy of geodesic flow on closed surfaces of
negative Euler characteristic with smooth non-positively curved Riemannian
metrics with fixed total area in a fixed conformal class. Moreover, we obtain a
collar lemma, a thick-thin decomposition, and precompactness for the considered
class of metrics. Also, we extend some of the results to metrics of fixed total
area in a fixed conformal class with no focal points and with some integral
bounds on the positive part of the Gaussian curvature.

Dynamical Systems;Differential Geometry