Regularity of Kobayashi metric

We review some recent results on existence and regularity of Monge-Ampère exhaustions on the smoothly bounded strongly pseudocon- vex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric κ on each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It in- cludes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.