Entropy rigidity and flexibility for suspension flows over Anosov diffeomorphisms

For any $C^{\mathrm{\infty }}$, area-preserving Anosov diffeomorphism $f$ of a surface, we show that a suspension flow over $f$ is $C^{\mathrm{\infty }}$-conjugate to a constant-time suspension flow of a hyperbolic automorphism of the two torus if and only if the volume measure is the measure with maximal entropy. We also show that the the metric entropy with respect to the volume measure and the topological entropy of suspension flow over Anosov diffeomorphisms on torus achieve all possible values. Our results fit into two programs related to entropy rigidity and flexibility of Anosov systems.