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

## TL;DR

This paper extends rigidity results for complex hyperbolic manifolds by establishing a gap theorem involving diastatic entropy and degree of maps between Kähler manifolds, using barycentre map techniques.

## Contribution

It introduces a new gap theorem relating diastatic entropy and degree for maps between Kähler manifolds, extending previous rigidity results.

## Key findings

- Proves a gap theorem linking diastatic entropy and map degree.
- Extends Besson-Courtois-Gallot techniques to the Kähler setting.
- Provides conditions under which complex hyperbolic manifolds exhibit rigidity.

## Abstract

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].