Diastatic entropy and rigidity of hyperbolic manifolds

TL;DR

This paper establishes a lower bound for the diastatic entropy of compact Kähler manifolds mapped to hyperbolic manifolds, leading to rigidity results when the bound is attained, similar to volume entropy cases.

Contribution

It provides the first lower bound for diastatic entropy in terms of map degree and hyperbolic entropy, with rigidity theorems for equality cases.

Findings

Lower bound for diastatic entropy in terms of target entropy and degree

Rigidity theorems when the lower bound is achieved

Characterization of minimal diastatic entropy as hyperbolic metric

Abstract

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 $g_0$.