The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the K\"ahler case. Our main question is the existence of almost K\"ahler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.

中文翻译：

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近厄米几何中的可积性定理和共形常数陈标量曲率度量

研究了几乎 Hermitian 流形上的各种标量曲率，特别是关于共形变化。我们展示了几个可积性定理，它们表明其中两个只能在 K\"ahler 情况下一致。我们的主要问题是几乎存在 K\"ahler 度量，其具有共形常数陈标量曲率。这个问题在规则流形和互补情况下完全解决，在这种情况下，Chern-Yamabe 问题的方法适用于不可积的情况。还给出了对问题的矩图解释，导致了 Futaki 不变量和几何不变量理论的通常图片。