Existence and geometry of Hermitian metrics with constant second scalar curvature

We study Hermitian metrics with constant second scalar curvature on compact manifolds. We first consider a Yamabe-type problem for the second Bismut scalar curvature under a natural topological condition, and then analyze elliptic equations arising from constant second Chern scalar curvature within a fixed Hermitian conformal class and derive geometric consequences. Finally, under an Einstein-type condition on the second Chern curvature, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, which in certain cases further implies the existence of a Kähler-Einstein metric.

💡 Research Summary

The paper investigates Hermitian metrics on compact complex manifolds whose second scalar curvature—associated either with the Bismut connection or with the Chern connection—remains constant. The authors treat three distinct but related problems: a Yamabe‑type problem for the second Bismut scalar curvature, the existence and uniqueness of metrics with constant second Chern scalar curvature within a fixed Hermitian conformal class, and rigidity results under an Einstein‑type condition involving the second Chern curvature.

1. A Yamabe‑type problem for the second Bismut scalar curvature.
The authors first consider the equation obtained by conformally changing a Hermitian metric (\omega) to (\tilde\omega=e^{2u/(n-1)}\omega). Under this change the second Bismut scalar curvature transforms as
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