Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds

We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order $d\in [1,+\mathrm{\infty }]$. Finally, based on these results, we focus on some open problems naturally arising.