Sala Seminari (Dip. Matematica).
The volume growth entropy is the exponential growth rate of volume of balls in a Riemannian manifold. A conjecture of Gromov and Katok in the early 80’s states that the only knowledge of the entropy gives important informations about the ambient geometry, for instance in the case of locally symmetric spaces. Precisely it states that the volume entropy is (uniquely) minimized (among all metrics of prescribed volume) at the symmetric metric. This is the so-called minimal entropy problem. I’ll give an overview of what is known for this problem as well as surprising consequences regarding the Gromov’s minimal volume or degree theory. I’ll give a full proof of the case of product of hyperbolic planes (2015) and, if time permits, a glimpse for the case of Siegel upper half plane (work in progress).