Extremal almost-K\"ahler metrics

TL;DR

This paper extends the concepts of Futaki invariant and extremal vector fields to almost-Kähler manifolds, providing explicit formulas and examples of extremal metrics that saturate known estimates.

Contribution

It generalizes key invariants and vector fields from Kähler to almost-Kähler geometry and constructs explicit extremal metrics with specific properties.

Findings

Periodic extremal vector field when symplectic form is integral

Explicit formula for hermitian scalar curvature in Darboux coordinates

Examples of non-integrable extremal almost-Kähler metrics saturating LeBrun's estimates

Abstract

We generalize the notions of the Futaki invariant and extremal vector field of a compact K\"ahler manifold to the general almost-Kahler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-K\"ahler metrics saturating LeBrun's estimates.