Electrostatic system with divergence-free Bach tensor and non-null cosmological constant
-
Benedito Leandro
and
Róbson Lousa
Abstract
We prove that three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat, provided that the electric field and the gradient of the lapse function are linearly dependent. Consequently, a three-dimensional electrostatic manifold admits a local warped product structure with a one-dimensional base and a constant curvature surface fiber.
Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Award Identifier / Grant number: 303157/2022-4
Award Identifier / Grant number: 403349/2021-4
Funding statement: Benedito Leandro was partially supported by CNPq/Brazil Grant 303157/2022-4. Róbson Lousa was partially supported by PROPG-CAPES [Finance Code 001]. The authors were partially supported by CNPq Grant 403349/2021-4.
Acknowledgements
The authors would like to express their gratitude to Professor Tiarlos Cruz for his valuable comments and insightful discussions. We also want to thank the referee for carefully reading this work and for the relevant remarks.
References
[1] R. Bach, Zur Weylschen Relativitatsttheorie, Math. Z. 9 (1921), 110–135. 10.1007/BF01378338Search in Google Scholar
[2] A. L. Besse, Einstein manifolds, Class. Math., Springer, Berlin, 2007. Search in Google Scholar
[3] H.-D. Cao and Q. Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), no. 6, 1149–1169. 10.1215/00127094-2147649Search in Google Scholar
[4] H.-D. Cao, X. Sun and Y. Zhang, On the structure of gradient Yamabe solitons, Math. Res. Lett. 19 (2012), no. 4, 767–774. 10.4310/MRL.2012.v19.n4.a3Search in Google Scholar
[5] G. Catino, P. Mastrolia and D. D. Monticelli, Gradient Ricci solitons with vanishing conditions on Weyl, J. Math. Pures Appl. (9) 108 (2017), no. 1, 1–13. 10.1016/j.matpur.2016.10.007Search in Google Scholar
[6] C. Cederbaum and G. J. Galloway, Uniqueness of photon spheres in electro-vacuum spacetimes, Classical Quantum Gravity 33 (2016), no. 7, Article ID 075006. 10.1088/0264-9381/33/7/075006Search in Google Scholar
[7] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci Flow, Grad. Stud. Math. 77, American Mathematical Society, Providence, 2006. Search in Google Scholar
[8] P. T. Chruściel and E. Delay, Non-singular, vacuum, stationary space-times with a negative cosmological constant, Ann. Henri Poincaré 8 (2007), no. 2, 219–239. 10.1007/s00023-006-0306-4Search in Google Scholar
[9] P. T. Chruściel and E. Delay, Non-singular space-times with a negative cosmological constant: II. Static solutions of the Einstein-Maxwell equations, Lett. Math. Phys. 107 (2017), no. 8, 1391–1407. 10.1007/s11005-017-0955-xSearch in Google Scholar PubMed PubMed Central
[10] T. Cruz, V. Lima and A. de Sousa, Min-max minimal surfaces, horizons and electrostatic systems, preprint (2019), https://arxiv.org/abs/1912.08600; to appear in J. Differential Geom. Search in Google Scholar
[11] S. Fernando, Born–Infeld–de Sitter gravity: cold, ultra-cold and Nariai black holes, Internat. J. Modern Phys. D 22 (2013), no. 13, Article ID 1350080. 10.1142/S0218271813500806Search in Google Scholar
[12] S. Hwang and G. Yun, Vacuum static spaces with vanishing of complete divergence of Weyl tensor, J. Geom. Anal. 31 (2021), no. 3, 3060–3084. 10.1007/s12220-020-00384-4Search in Google Scholar
[13] H. K. Kunduri and J. Lucietti, No static bubbling spacetimes in higher dimensional Einstein–Maxwell theory, Classical and Quantum Gravity 35 (2018), no. 5, 054003. 10.1088/1361-6382/aaa744Search in Google Scholar
[14] B. Leandro, Vanishing conditions on Weyl tensor for Einstein-type manifolds, Pacific J. Math. 314 (2021), no. 1, 99–113. 10.2140/pjm.2021.314.99Search in Google Scholar
[15] B. Leandro, M. Andrade and R. Lousa, On the geometry of electrovacuum spaces in higher dimensions, Ann. Henri Poincaré 24 (2023), no. 9, 3153–3184. 10.1007/s00023-023-01306-0Search in Google Scholar
[16] J. Lucietti, All higher-dimensional Majumdar–Papapetrou black holes, Ann. Henri Poincaré 22 (2021), no. 7, 2437–2450. 10.1007/s00023-021-01037-0Search in Google Scholar
[17] J. Qing and W. Yuan, A note on static spaces and related problems, J. Geom. Phys. 74 (2013), 18–27. 10.1016/j.geomphys.2013.07.003Search in Google Scholar
[18] D. C. Robinson, A simple proof of the generalization of Israel’s theorem, Gen. Relativ. Gravit. 8 (1977), 695–698. 10.1007/BF00756322Search in Google Scholar
[19] P. Szekeres, Conformal tensors, Proc. Roy. Soc. Lond. Ser. A. Math. Phys. Sci. 304 (1968), no. 1476, 113–122. 10.1098/rspa.1968.0076Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Open orbits and primitive zero ideals for solvable Lie algebras
- On the Pauli group on 2-qubits in dynamical systems with pseudofermions
- Electrostatic system with divergence-free Bach tensor and non-null cosmological constant
- Perturbation of domain for the linear parabolic equation
- K-theory of flag Bott manifolds
- Some results on Seshadri constants of vector bundles
- Strichartz inequality for orthonormal functions associated with special Hermite operator
- The globally smooth solutions and asymptotic behavior of the nonlinear wave equations in dimension one with multiple speeds
- On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems
- Boundedness of commutators of rough Hardy operators on grand variable Herz spaces
- Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets
- An alternative proof of Tataru’s dispersive estimates
- The p-Bohr radius for vector-valued holomorphic and pluriharmonic functions
- Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction
- Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case
- A class of quaternionic Fourier orthonormal bases
- Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields
- Normalized solutions for scalar field equation involving multiple critical nonlinearities
Articles in the same Issue
- Frontmatter
- Open orbits and primitive zero ideals for solvable Lie algebras
- On the Pauli group on 2-qubits in dynamical systems with pseudofermions
- Electrostatic system with divergence-free Bach tensor and non-null cosmological constant
- Perturbation of domain for the linear parabolic equation
- K-theory of flag Bott manifolds
- Some results on Seshadri constants of vector bundles
- Strichartz inequality for orthonormal functions associated with special Hermite operator
- The globally smooth solutions and asymptotic behavior of the nonlinear wave equations in dimension one with multiple speeds
- On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems
- Boundedness of commutators of rough Hardy operators on grand variable Herz spaces
- Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets
- An alternative proof of Tataru’s dispersive estimates
- The p-Bohr radius for vector-valued holomorphic and pluriharmonic functions
- Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction
- Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case
- A class of quaternionic Fourier orthonormal bases
- Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields
- Normalized solutions for scalar field equation involving multiple critical nonlinearities
