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.