On a variational theorem of Gauduchon and torsion-critical manifolds

Abstract

In 1984, Gauduchon [Math. Ann. 267 (1984), pp. 495–518] considered the functional of L 2 L^2 -norm of his torsion 1 1 -form on a compact Hermitian manifold. He obtained the Euler-Lagrange equation for this functional, and showed that in dimension 2 2 the critical metrics must be balanced (namely with vanishing torsion 1 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 L^2 -norm of the full Chern torsion, and show by examples that there are critical points of this functional that are not Kähler.