Diastatic entropy and rigidity of hyperbolic manifolds

Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic K\"ahler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary, when $X=Y$, we show that the minimal diastatic entropy is achieved if and only if $g$ is holomorphically or anti-holomorphically isometric to the hyperbolic metric $g_0$

The volume entropy of local Hermitian symmetric space of noncompact type

We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant $r$, $a$, $b$.Comment: 10 page

On the diastatic entropy and C^1-rigidity of complex hyperbolic manifolds

Let f:(Y,g)->(X,g_0) be a non zero degree continuous map between compact K\"ahler manifolds of dimension greater or equal to 2, where g_0 has constant negative holomorphic sectional curvature. Adapting the Besson-Courtois-Gallot barycentre map techniques to the K\"ahler setting, we prove a gap theorem in terms of the degree of f and the diastatic entropies of (Y, g) and (X,g_0), which extends the rigidity result proved by the author in [13].Comment: 23 page

A note on diastatic entropy and balanced metrics

We give un upper bound Ent(\Omega, g)<\lambda\ of the diastatic entropy Ent(\Omega, g) of a complex bounded domain (\Omega, g) in terms of the balanced condition (in Donaldson terminology) of the Kaehler metric \lambda g. When (\Omega, g) is a homogeneous bounded domain we show that the converse holds true, namely if Ent(\Omega, g)<1 then g is balanced. Moreover, we explcit compute Ent(\Omega, g) in terms of Piatetski-Shapiro constants.Comment: 7 page

Some remarks on homogeneous K\"ahler manifolds

In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous K\"ahler manifold M endowed with an integral K\"ahler form $\mu\omega$, admits a holomorphic isometric immersion in the complex projective space, for a suitable $\mu>0$. This result has two corollaries which extend to homogeneous K\"ahler manifolds the results obtained by the authors in [8] and in [12] for homogeneous bounded domains.Comment: 8 page