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].