Finsler metrics of complex domains with Monge-Ampere exhaustions

speaker: Andrea Spiro (University of Camerino)

abstract: A smooth Monge-Ampère exhaustion on a smoothly bounded strongly pseudoconvex domain \(D \subset {\mathbb C}^n\) is a continuous strictly plurisubharmonic exhaustion \(\tau: \overline D \to [0,1]\), which is \(\mathcal C^\infty\) at each point different from the unique minimum point \(z\) and such that \(u := \log \tau\) satisfies the homogeneous complex Monge-Ampère equation \( (d d^c u)^n = 0\). The class of domains admitting at least one such exhaustion includes all strictly convex domains with smooth boundaries and many others. In this talk we present a recent result with Giorgio Patrizio on existence of an infinite number of smooth Monge-Ampère exhaustions on each smooth domain of \(\mathbb C^n\) that admits at least one Monge-Ampère exhaustion - namely there exists at least one such exhaustion per each point of the domain. This implies that the Kobayashi pseudo-metric of any such domain is actually a smooth complex Finsler metric with very special differential geometric properties.

timetable:
Wed 23 May, 10:40 - 11:30, Aula Dini
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