A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the \(L^2\)-norm of the Weyl curvature. When the Riemannian 4-manifold in question is a Kähler surface, we provide a rough classification of solutions, followed by detailed results regarding each case in the classification. The most mysterious case prominently involves 3-dimensional CR manifolds.

中文翻译：

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巴赫平Kähler表面

如果紧凑4流形上的黎曼度量是Weyl曲率\（L ^ 2 \）-范数的临界点，则称其为Bach平坦。当所讨论的黎曼4流形为Kähler曲面时，我们提供解决方案的粗略分类，然后提供有关分类中每种情况的详细结果。最神秘的案例主要涉及3维CR流形。