# Variational problems in conformal geometry

## TL;DR

This paper investigates extremal metrics in conformal classes of almost Hermitian metrics, identifying unique representatives that minimize certain functionals involving the Lee form, extending known results from complex surfaces to higher dimensions.

## Contribution

It introduces a new class of extremal metrics in conformal geometry, extending the concept of Gauduchon metrics to higher dimensions and establishing their uniqueness.

## Key findings

- Gauduchon metrics are extremal for a specific functional involving the Lee form.
- Existence of unique extremal metrics in each conformal class on compact complex surfaces.
- Extension of extremal metric concept to higher dimensions with uniqueness results.

## Abstract

We study the Euler-Lagrange equation for several natural functionals defined on a conformal class of almost Hermitian metrics, whose expression involves the Lee form $\theta$ of the metric. We show that the Gauduchon metrics are the unique extremal metrics of the functional corresponding to the norm of the codifferential of the Lee form. We prove that on compact complex surfaces, in every conformal class there exists a unique metric, up to multiplication by a constant, which is extremal for the functional given by the $L^2$-norm of $dJ\theta$, where $J$ denotes the complex structure. These extremal metrics are not the Gauduchon metrics in general, hence we extend their definition to any dimension and show that they give unique representatives, up to constant multiples, of any conformal class of almost Hermitian metrics.