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Mean-stable surfaces in static Einstein–Maxwell theory

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Abstract

We use the theory of mean-stable surfaces (stable minimal surfaces included) to explore the static Einstein–Maxwell space-time. We first prove that the zero set of the lapse function must be contained in the horizon boundary. Then, we explore some implications of it providing some results of the nonexistence of stable minimal surfaces in the interior of an electrostatic space, subject to certain initial-boundary data. We finish by proving that the ADM mass is bounded from above by the Hawking quasi-local mass with charge.

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References

  1. Anderson, M.T.: On the structure of solutions to the static vacuum Einstein equations. Ann. Henri Poincaré 1(6), 995–1042 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bartnik, R.: New definition of quasilocal mass. Phys. Rev. Lett. 62(20), 2346–2348 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bethuel, F., et al.: Calculus of Variations and Geometric Evolution Problems. Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held in Cetaro, Italy, June 15-22, 1996. Springer (2006)

  4. Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burko, L.M., Khanna, G., Sabharwal, S.: Scalar and gravitational hair for extreme Kerr black holes. Phys. Rev. D 103(2), L021502 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  6. Cai, M.: Galloway. Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature. Commun. Anal. Geom. 8(3), 565–573 (2000)

    Article  MATH  Google Scholar 

  7. Cao, H.-D., Chen, Q.: On Bach-flat gradient shrinking Ricci solitons. Duke Math. J. 162(6), 1149–1169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cederbaum, C., Galloway, G.J.: Uniqueness of photon spheres via positive mass rigidity. Commun. Anal. Geom. 25(2), 303–320 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chruściel, P.T.: Towards a classification of static electrovacuum spacetimes containing an asymptotically flat spacelike hypersurface with compact interior. Class. Quant. Gravity 16(3), 689–704 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Christodoulou, D., Yau, S.-T.: Some remarks on the quasi-local mass. Mathematics and general relativity (Santa Cruz, CA, 1986), 9-14, Contemp. Math., 71, Amer. Math. Soc., Providence, RI (1988)

  11. Claudel, C.-M., Virbhadra, K.S., Ellis, G.F.R.: The geometry of photon surfaces. J. Math. Phys. 42(2), 818–838 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Cruz, T., Lima, V., de Sousa, A.: Min-max minimal surfaces, horizons and electrostatic systems. to appear in JDG. arXiv:1912.08600

  13. Galloway, G.J.: On the topology of black holes. Commun. Math. Phys. 151(1), 53–66 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Hartle, J.B., Hawking, S.W.: Solutions of the Einstein-Maxwell equations with many black holes. Commun. Math. Phys. 26, 87–101 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  15. Huang, L.-H., Martin, D., Miao, P.: Static potentials and area minimizing hypersurfaces. Proc. Am. Math. Soc. 146(6), 2647–2661 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Horowitz, Gary.: The Dark Side of String Theory: Black Holes and Black Strings. String theory and quantum gravity ’91. Proceedings of the 1991 Trieste Spring School and Workshop held April 15–26, (1991). Edited by J. Harvey, R. Iengo, K. S. Narain, S. Randjbar-Daemi and H. Verlinde. World Scientific Publishing Co., Inc., River Edge, NJ, 1992. x+383 pp. ISBN: 981-02-0774-3 81-06 (81T40)

  17. Jahns, S.: Photon sphere uniqueness in higher-dimensional electrovacuum spacetimes. Classif. Quant. Gravity 36(23), 235019, 24 pp (2019)

    MathSciNet  MATH  Google Scholar 

  18. Khuri, M.: A Penrose-like inequality with charge. Gen. Relativ. Gravit. 45, 2341–2361 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Khuri, M., Weinstein, G., Yamada, S.: - Proof of the Riemannian Penrose inequality with charge for muktiple black holes. J. Differ. Geom. 106, 451–498 (2017)

    Article  MATH  Google Scholar 

  20. Kunduri, H.K., Lucietti, J.: No static bubbling spacetimes in higher dimensional einstein-maxwell theory. Class. Quant. Gravity 35(5), p.054003 (9pp) (2018)

    Article  ADS  MATH  Google Scholar 

  21. Liu, G.: \(3\)-manifolds with nonnegative Ricci curvature. Invent. Math. 193(2), 367–375 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Li, P., Yau, S.-T.: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69(2), 269–291 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Miao, P.: On existence of static metric extensions in general relativity. Commun. Math. Phys. 241(1), 27–46 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Miao, P.: A remark on boundary effects in static vacuum initial data sets. Class. Quant. Gravity 22(11), L53–L59 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Miao, P., Tam, L.-F.: Static potentials on asymptotically flat manifolds. Ann. Henry Poincaré 16(10), 2239–2264 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Ruback, P.: A new uniqueness theorem for charged black holes. Class. Quant. Gravity 5(10), L155–L159 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. IME, Universidade Federal de Goiás, Goiânia, GO, CEP 74690-900, Brazil

    Fernando Coutinho & Benedito Leandro

  2. CEST, Universidade do Estado do Amazonas, 1085, Tefé, AM, CEP 69552-315, Brazil

    Fernando Coutinho

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  1. Fernando Coutinho
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  2. Benedito Leandro
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Correspondence to Benedito Leandro.

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Fernando Soares Coutinho was partially supported by PROPG-CAPES/FAPEAM. Benedito Leandro was partially supported by CNPq-Grant 403349/2021-4.

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Coutinho, F., Leandro, B. Mean-stable surfaces in static Einstein–Maxwell theory. Lett Math Phys 112, 128 (2022). https://doi.org/10.1007/s11005-022-01623-1

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