On the diastatic entropy and [formula omitted]-rigidity of complex hyperbolic manifolds

Abstract

Let f:(Y,g)→(X,g0) be a nonzero degree continuous map between compact Kähler manifolds of dimension n≥2, where g0 has constant negative holomorphic sectional curvature. Adapting the Besson–Courtois–Gallot barycentre map techniques to the Kähler setting, we prove a gap theorem in terms of the degree of f and the diastatic entropies of (Y,g) and (X,g0) which extends the rigidity result proved by the author in [13].