# Propagation of regularity for Monge-Ampere exhaustions and Kobayashi metrics

##
Andrea Spiro
(Università di Camerino)

created by risa on 18 Apr 2018

24 apr 2018
-- 16:00

Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma

**Abstract.**

A Monge-Ampère exhaustion of class $C^\infty$ on a smoothly bounded strongly
pseudoconvex domain $D \subset {\bf C}^n$ is a continuous strictly plurisubharmonic
exhaustion $\tau: \overline D \to [0,1]$, which is $C^\infty$ at each point different from
the unique minimum point $z$ and such that $u := \log \tau$ satisfies the homogeneous
complex Monge-Ampère equation $(d d^c u)^n = 0$.
Domains admitting one such exhaustion are called {\it smooth domains of circular type}.
They includes all strongly pseudoconvex circular domains and all strictly
convex domains with smooth boundaries.
In this talk we present a recent result with Giorgio Patrizio on the existence of an infinite
family of Monge-Ampère exhaustions of class $C^\infty$
on any smooth domain of circular type, namely, at least one per each point of the domain.
This implies that the Kobayashi pseudo-metric of a smooth domain of circular type is actually a
smooth strongly pseudoconvex Finsler metric.