Regularity of Kobayashi metric

TL;DR

This paper reviews recent advances in the regularity of Monge-Ampère exhaustions on strongly pseudoconvex domains, showing that the Kobayashi pseudo-metric becomes a smooth Finsler metric under certain conditions.

Contribution

It establishes that on a broad class of domains, the Kobayashi pseudo-metric is actually smooth, extending previous results to include various deformations of circular domains.

Findings

Kobayashi pseudo-metric is smooth Finsler metric on certain domains.

Regularity of Monge-Ampère exhaustions is crucial for metric smoothness.

Results apply to all smoothly bounded strongly pseudoconvex complete circular domains and their small deformations.

Abstract

We review some recent results on existence and regularity of Monge-Amp\`ere exhaustions on the smoothly bounded strongly pseudoconvex 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 k on an appropriare open subset of each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.