On a variational theorem of Gauduchon and torsion-critical manifolds

Dongmei Zhang; Fangyang Zheng

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arXiv · 2205.12480

On a variational theorem of Gauduchon and torsion-critical manifolds

Dongmei Zhang·Fangyang Zheng

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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.

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Dongmei Zhang, Fangyang Zheng. 2022-05-25. On a variational theorem of Gauduchon and torsion-critical manifolds. https://arxiv.org/abs/2205.12480

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