Stationary disks and Green functions in almost complex domains

TL;DR

This paper develops normal forms for almost complex domains with stationary disks, enabling the construction of counterexamples and the analysis of extremal properties related to the Kobayashi metric and Monge-Ampere equations.

Contribution

It introduces generalized Riemann maps and normal forms for almost complex domains, advancing the understanding of stationary disks and their extremal properties.

Findings

Normal forms for almost complex domains are established.

Counterexamples are constructed using these normal forms.

Conditions for stationary disks to be extremal for the Kobayashi metric are identified.

Abstract

Using generalized Riemann maps, normal forms for almost complex domains (D, J) with singular foliations by stationary disks are defined. Such normal forms are used to construct counterexamples and to determine intrinsic conditions, under which the stationary disks are extremal disks for the Kobayashi metric or determine solutions to almost complex Monge-Ampere equation.