Annales Henri Poincaré

, Volume 14, Issue 7, pp 1747–1773 | Cite as

Convexity of Reduced Energy and Mass Angular Momentum Inequalities

  • Richard Schoen
  • Xin Zhou


In this paper, we extend the work in Chruściel and Costa (Class. Quant. Grav. 26:235013, 2009), Chruściel et al. (Ann. Phy. 323:2591–2613, 2008), Costa (J. Math. Theor. 43:285202, 2010), Dain (J. Diff. Geom. 79:33–67, 2008). We weaken the asymptotic conditions on the second fundamental form, and we also give an L 6−norm bound for the difference between general data and Extreme Kerr data or Extreme Kerr–Newman data by proving convexity of the renormalized Dirichlet energy when the target has non-positive curvature. In particular, we give the first proof of the strict mass/angular momentum/charge inequality for axisymmetric Einstein/Maxwell data which is not identical with the extreme Kerr–Newman solution.


Black Hole Asymptotic Condition Dirichlet Energy Positive Mass Theorem Black Hole Case 
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  1. 1.
    Brill D.: On the positive definite mass of the Bondi-Weber-Wheeler time- symmetric gravitational waves. Ann. Phys. 7, 466–483 (1959)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Chruściel P.T.: Mass and Angular-Momentum Inequalities for Axi-Symmetric Initial Data Sets. I. Positivity of Mass. Ann. Phys. 323, 2566–2590 (2008)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Chruściel P.T., Costa J.L.: Mass, angular-momentum and charge inequalities for axisymmetric initial data. Class. Quant. Grav. 26, 235013 (2009)ADSCrossRefGoogle Scholar
  4. 4.
    Chruściel P.T., Li Y., Weinstein G.: Mass and angular-momentum inequalities for axi-symmetric initial data sets. II. angular momentum. Ann. Phys. 323, 2591–2613 (2008)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Chruściel P.T., Nguyen L.: A uniqueness theorem for degenerate Kerr-Newman black holes. Ann. Henri Poincaré 11, 585–609 (2010)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Costa J.L.: Proof of a Dain inequality with charge. J. Phys. A: Math. Theor. 43, 285202 (2010)CrossRefGoogle Scholar
  7. 7.
    Dain S.: Proof of the angular momentum-mass inequality for axisymmetric black hole. J. Differential Geom. 79, 33–67 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dain S.: A variational principle for stationary, axisymmetric solutions of Einstein’s equations. Class. Quant. Grav. 23, 6857–6871 (2006)MathSciNetADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Evans L.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)Google Scholar
  10. 10.
    Schoen, R.: Analytic Aspect of Harmonic Maps. In: ChernSeminar, S.S. (ed.) on Nonlinear PDE, MSRI Publication, pp. 321–358, Springer-Verlag, New York (1984)Google Scholar
  11. 11.
    Weinstein G.: N-black hole stationary and axially symmetric solutions of the Einstein/Maxwell equations. Commun. Part. Diff. Eqs. 21, 1389–1430 (1996)CrossRefzbMATHGoogle Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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