Chern flat manifolds that are torsion-critical
Dongmei Zhang, Fangyang Zheng
TL;DR
The paper classifies torsion-critical metrics on compact Chern flat manifolds by analyzing the universal cover G, a complex Lie group. It proves that non-Kähler torsion-critical metrics force G to be semi-simple, and that every semi-simple G admits a canonical left-invariant metric g0 that is torsion-critical; the authors also give explicit structure constants leading to A = B = I and T^j_{ik} = -\sqrt{2}\,S^j_{ik} for g0. A key open question remains whether all torsion-critical metrics on a semi-simple G are proportional to the canonical metric or decompose along simple factors. The work connects non-Kähler torsion-critical geometry with Lie-theoretic structure and provides a complete classification in the Chern-flat setting.
Abstract
In our previous work, we introduced a special type of Hermitian metrics called {\em 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.