# Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry

## TL;DR

This paper investigates scalar curvatures on almost Hermitian manifolds, establishing integrability conditions, and explores the existence of conformally constant Chern scalar curvature metrics, providing solutions for specific manifold classes and a moment map framework.

## Contribution

It proves integrability theorems linking scalar curvatures to the Kähler condition and solves the conformal scalar curvature problem for ruled and certain non-integrable manifolds.

## Key findings

- Two scalar curvatures agree only in the Kähler case.
- Complete solution for conformally constant Chern scalar curvature on ruled manifolds.
- Development of a moment map and Futaki invariant framework.

## Abstract

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the K\"ahler case. Our main question is the existence of almost K\"ahler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.