**Abstract**

It is a classic problem to determine when a continuous application between two smooth and closed manifolds is homotopic to a more regular one. The most important example is the seminal Mostow's rigidity theorem, which states that if two locally symmetric compact manifolds with strictly negative curvature are homotopically equivalent, then they are isometric up to an homothetic factor. The Mostow theorem was wonderfully extended by G. Besson, G. Courtois and S. Gallot. Given two Riemannian manifolds (Y, g) and (X, g0) of dimension e 3, with g0 of strictly negative sectional curvature, they provide naturals sufficient conditions, in terms of volume entropy, in order that a continuous non-zero degree map f: (Y, g) -> (X, g0) is homotopic to a Riemannian covering F: (Y, g) -> (X, g0). As an application of this result and the techniques developed, in particular the barycentre map technique, they were able to solve a series of longstanding problems. The key tool of this project is the "diastatic entropy", a new Kaehlerian invariant that I defined in the paper "A note on diastatic entropy and balanced metrics", J. Geom. Phys. 2014. This invariant has some properties analogous to the "volume entropy", but it is surprisingly linked to the balance condition (in the sense of S. Donaldson) and to the Berezin quantization of the involved manifold. In addition, its properties allowed me to prove the complex version of the Mostow rigidity theorems and to given a characterization of the hyperbolic metric as the metric that realizes the minimal diastatic entropy. ("Diastatic entropy and rigidity of complex hyperbolic manifolds", Complex Manifolds 3. (2016), 186-192). The definition of diastatic entropy is in terms of the diastasis function of Calabi, an object that determines the geometry of a variety of Kaehler and which (contrary to the distance function) respects the submanifolds. This property has been proved extremely useful in the past to study the rigidity of a variety of Kaehler (see for example works done by E. Calabi, A. Loi, N. Mok or M. Umehara). Therefore, as a consequence of the properties of the diastasis function and the results already obtained on the diastatic entropy, I believe that a study of this new Kaehlerian invariant (the diastatic entropy), with techniques similar to those used by Besson-Courtois-Gallot for the study of the volume entropy (in particular adapting the barycentre map technique), could be a new field of extremely fruitful research. (AU)