On dynamically convex contact manifolds and filtered symplectic homology

In this paper, we are interested in characterizing the standard contact sphere in terms of dynamically convex contact manifolds, which admit a Liouville filling with vanishing symplectic homology. We first observe that if the filling is flexible, then those contact manifolds are contactomorphic to the standard contact sphere. We then investigate quantitative geometry of those contact manifolds focusing on similarities with the standard contact sphere in filtered symplectic homology.