Foliations by stationary disks of almost complex domains

TL;DR

This paper investigates the existence of stationary disks in almost complex domains, showing that small deformations of strictly linearly convex domains admit foliations by stationary disks through any internal or boundary point.

Contribution

It demonstrates that small deformations of strictly linearly convex domains in complex space admit foliations by stationary disks, extending classical results to almost complex structures.

Findings

Existence of stationary disk foliations in deformed domains

Foliations passing through any internal point

Foliations passing through a boundary point

Abstract

We study the problem of existence of stationary disks for domains in almost complex manifolds. As a consequence of our results, we prove that any almost complex domains which is a small deformations of a strictly linearly convex domain $D \subset C^n$ with standard complex structure admits a singular foliation by stationary disks passing through any given internal point. Similar results are given for foliation by stationary disks through a given boundary point