Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold with Boundary

Evangelista, Israel; Freitas, A. A.; Viana, Emanuel

doi:10.1007/s00025-022-01749-7

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2023

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“…In this case, the function ψ is called conformal factor of V . Riemannian manifolds admitting conformal vector fields has been extensively studied in [3,4,5,6,7,8,13,17,19,20] and references are therein.…”

Section: Introductionmentioning

confidence: 99%

Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor

Hwang

Yun

2020

J Geom Anal

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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 Bach tensor and Weyl 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. On the other hand, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.

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