The diastasis function

In the first part of the seminar, I will introduce Calabi's {diastasis function}, a special Kähler potential defined on real analytic Kähler manifold. Then I will illustrate some of its applications. In the second part, I will apply the properties of the diastasis function to study the problem of Minimal Entropy on a compact Riemannian manifold $(X,g_0)$ which is a quotient of the polydisc $(\mathbb C H^1)^r$ endowed with its canonical K\{a}hler structure. Namely