Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Weyl tensor with the non-negativity of the complete divergence of the Bach tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. Second, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness of the metric. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.