On a variational theorem of Gauduchon and torsion-critical manifolds

TL;DR

This paper extends Gauduchon's variational theorem from 2-dimensional Hermitian manifolds to higher dimensions, demonstrating that critical metrics are balanced and exploring the nature of critical points related to the Chern torsion.

Contribution

It generalizes Gauduchon's result to all dimensions and investigates critical points of the torsion functional beyond Kähler metrics.

Findings

Critical metrics are balanced in all dimensions.

Existence of critical points of the torsion functional that are not Kähler.

Extension of the variational theorem to higher-dimensional Hermitian manifolds.

Abstract

In 1984, Gauduchon considered the functional of $L^2$-norm of his torsion $1$-form on a compact Hermitian manifold. He obtained the Euler-Lagrange equation for this functional, and showed that in dimension $2$ the critical metrics must be balanced (namely with vanishing torsion $1$-form). In this note we extend his result to higher dimensions, and show that critical metrics are balanced in all dimensions. We also consider the $L^2$-norm of the full Chern torsion, and show by examples that there are critical points of this functional that are not K\"ahler.