Chern flat manifolds that are torsion-critical

TL;DR

This paper characterizes Chern flat manifolds that are torsion-critical, showing that such metrics exist precisely on semi-simple complex Lie groups and are critical points of the Chern torsion norm.

Contribution

It establishes a complete characterization of torsion-critical Chern flat metrics on compact quotients of complex Lie groups, linking them to semi-simple structures.

Findings

Torsion-critical Chern flat metrics occur only on semi-simple complex Lie groups.

Any semi-simple complex Lie group admits a torsion-critical Chern flat metric.

The paper provides a classification of torsion-critical metrics in this setting.

Abstract

In our previous work, we introduced a special type of Hermitian metrics called {\em torsion-critical,} which are non-K\"ahler 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.