Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 005, 15 pages      arXiv:2207.14563
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

On Asymptotically Locally Hyperbolic Metrics with Negative Mass

Piotr T. Chruściel a and Erwann Delay b
a)  Faculty of Physics, University of Vienna, Boltzmanngasse 5, A 1090 Vienna, Austria
b)  Laboratoire de Mathématiques d'Avignon, Avignon Université, F-84916 Avignon and F.R.U.M.A.M., CNRS, F-13331 Marseille, France

Received August 01, 2022, in final form January 17, 2023; Published online January 23, 2023

We construct families of asymptotically locally hyperbolic Riemannian metrics with constant scalar curvature (i.e., time symmetric vacuum general relativistic initial data sets with negative cosmological constant), with prescribed topology of apparent horizons and of the conformal boundary at infinity, and with controlled mass. In particular we obtain new classes of solutions with negative mass.

Key words: scalar curvature; asymptotically hyperbolic manifolds; negative mass.

pdf (463 kb)   tex (84 kb)  


  1. Barzegar H., Chruściel P.T., Hörzinger M., Energy in higher-dimensional spacetimes, Phys. Rev. D 96 (2017), 124002, 25 pages, arXiv:1708.03122.
  2. Birmingham D., Topological black holes in anti-de Sitter space, Classical Quantum Gravity 16 (1999), 1197-1205, arXiv:hep-th/9808032.
  3. Chruściel P.T., Geometry of black holes, Internat. Ser. Monogr. Phys., Vol. 169, Oxford University Press, Oxford, 2020.
  4. Chruściel P.T., Delay E., Exotic hyperbolic gluings, J. Differential Geom. 108 (2018), 243-293, arXiv:1511.07858.
  5. Chruściel P.T., Delay E., The hyperbolic positive energy theorem, arXiv:1901.05263.
  6. Chruściel P.T., Delay E., Wutte R., Hyperbolic energy and Maskit gluings, arXiv:2112.00095.
  7. Galloway G.J., Schleich K., Witt D.M., Woolgar E., Topological censorship and higher genus black holes, Phys. Rev. D 60 (1999), 104039, 11 pages, arXiv:gr-qc/9902061.
  8. Guillarmou C., Moroianu S., Rochon F., Renormalized volume on the Teichmüller space of punctured surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), 323-384, arXiv:1504.04721.
  9. Herzlich M., Computing asymptotic invariants with the Ricci tensor on asymptotically flat and asymptotically hyperbolic manifolds, Ann. Henri Poincaré 17 (2016), 3605-3617, arXiv:1503.00508.
  10. Horowitz G.T., Myers R.C., AdS-CFT correspondence and a new positive energy conjecture for general relativity, Phys. Rev. D 59 (1999), 026005, 12 pages, arXiv:hep-th/9808079.
  11. Isenberg J., Lee J.M., Stavrov Allen I., Asymptotic gluing of asymptotically hyperbolic solutions to the Einstein constraint equations, Ann. Henri Poincaré 11 (2010), 881-927, arXiv:0910.1875.
  12. Kottler F., Über die physikalischen Grundlagen der Einsteinschen Gravitationstheorie, Ann. Phys. 56 (1918), 401-462.
  13. Mazzeo R., Pacard F., Maskit combinations of Poincaré-Einstein metrics, Adv. Math. 204 (2006), 379-412, arXiv:math.DG/0211099.
  14. Mazzeo R., Taylor M., Curvature and uniformization, Israel J. Math. 130 (2002), 323-346, arXiv:math.DG/0105016.
  15. Rupflin M., Hyperbolic metrics on surfaces with boundary, J. Geom. Anal. 31 (2021), 3117-3136, arXiv:1807.04464.
  16. Rupflin M., Topping P.M., Zhu M., Asymptotics of the Teichmüller harmonic map flow, Adv. Math. 244 (2013), 874-893, arXiv:1209.3783.
  17. Zhang T., Asymptotic properties of the hyperbolic metric on the sphere with three conical singularities, Bull. Korean Math. Soc. 51 (2014), 1485-1502, arXiv:1301.7272.

Previous article  Next article  Contents of Volume 19 (2023)