Stability under deformations of extremal almost-K\"ahler metrics in dimension 4

TL;DR

This paper investigates the stability of extremal almost-Kähler metrics in four dimensions, demonstrating that under certain conditions, such metrics persist under small deformations of the complex structure.

Contribution

It proves the existence of a smooth family of extremal almost-Kähler metrics under deformations, extending stability results in the context of 4-dimensional symplectic geometry.

Findings

Existence of a smooth family of extremal almost-Kähler metrics under short-time deformations.

Stability of extremal almost-Kähler metrics under certain hypotheses.

Persistence of the induced almost-complex structure up to diffeomorphism.

Abstract

Given a path of almost-K\"ahler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-K\"ahler metric is an extremal K\"ahler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-K\"ahler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.