Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor
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Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor Seungsu Hwang1 · Gabjin Yun2 Received: 29 May 2019 © Mathematica Josephina, Inc. 2020
Abstract 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. Keywords Vacuum static space · Bach tensor · Weyl tensor · Black holes · Besse conjecture · Einstein metric Mathematics Subject Classification 53C25 · 58E11
The first author was supported by the National Research Foundation of Korea (NRF-2018R1D1A1B05042186) and the second author was supported by the National Research Foundation of Korea (NRF-2019R1A2C1004948).
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Gabjin Yun [email protected] Seungsu Hwang [email protected]
1
Department of Mathematics, Chung-Ang University, 84 HeukSeok-ro, Dongjak-gu, Seoul 06974, Republic of Korea
2
Department of Mathematics, Myong Ji University, 116 Myongji-ro Cheoin-gu, Yongin, Gyeonggi 17058, Republic of Korea
123
S. Hwang, G. Yun
1 Introduction An n-dimensional complete Riemannian manifold (M, g) is said to be a static space with a perfect fluid if there exists a smooth non-trivial function f on M satisfying Dg d f − r g −
sg g n−1
f =
1 n
sg f + Δg f n−1
g,
(1.1)
where Dg d f is the Hessian of f , r g is the Ricci tensor of g with its scalar curvature sg , and Δg f is the (negative) Laplacian of f . In particular, if Δg f = −
sg f, n−1
(1.2)
(M, g) is said to be a vacuum static space. In this case, Eq. (1.1) reduces to Dg d f − r g −
sg g n−1
f = 0.
(1.3)
The above equation was considered by Fischer and Marsden [9] in their study of the surjectivity of a linearized scalar curvature functional in the space of Riemannian metrics (cf. [11,18]). More precisely, the linearized scalar curvature sg is given by sg (η) = −Δg tr g η + δδη − r g , η for any symmetric bilinear form η on M (cf. Here, δ = −div is the (negative) [3]). n D Ei η(E i , X ) for any vector X and divergence, which is defined by δη(X ) = − i=1 a local frame {E i }. Then, the L 2 -adjoint operator sg∗ with respect to the metric g is given by sg∗ ( f ) = Dg d f − (Δg f )g − f r g for any smooth function f on M. Thus, if a smooth function f on M is a solution of the vacuum static equation (1.3), then f is an element of the kernel space, ker sg∗ , of the operator sg∗ . By taking the divergence of (1.3), we have 21 f dsg = 0, which implies that sg is constant on M since the
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