Chern flat manifolds that are torsion-critical

Zhang, Dongmei; Zheng, Fangyang

doi:10.1090/proc/17135

Chern flat manifolds that are torsion-critical
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by Dongmei Zhang and Fangyang Zheng;

Proc. Amer. Math. Soc. 153 (2025), 1275-1282

DOI: https://doi.org/10.1090/proc/17135

Published electronically: January 29, 2025

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Abstract:

In our previous work, we introduced a special type of Hermitian metrics called torsion-critical, which are non-Kähler critical points of the $L^2$-norm of Chern torsion over the space of all Hermitian metrics with unit volume on a compact complex manifold. In this short note, we restrict our attention to the class of compact Chern flat manifolds, which are compact quotients of complex Lie groups equipped with compatible left-invariant metrics. Our main result states that, if a Chern flat metric is torsion-critical, then the complex Lie group must be semi-simple, and conversely, any semi-simple complex Lie group admits a compatible left-invariant metric that is torsion-critical.

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