Abstract
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.
Article PDF
Similar content being viewed by others
References
Brill D.: On the positive definite mass of the Bondi-Weber-Wheeler time- symmetric gravitational waves. Ann. Phys. 7, 466–483 (1959)
Chruściel P.T.: Mass and Angular-Momentum Inequalities for Axi-Symmetric Initial Data Sets. I. Positivity of Mass. Ann. Phys. 323, 2566–2590 (2008)
Chruściel P.T., Costa J.L.: Mass, angular-momentum and charge inequalities for axisymmetric initial data. Class. Quant. Grav. 26, 235013 (2009)
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)
Chruściel P.T., Nguyen L.: A uniqueness theorem for degenerate Kerr-Newman black holes. Ann. Henri Poincaré 11, 585–609 (2010)
Costa J.L.: Proof of a Dain inequality with charge. J. Phys. A: Math. Theor. 43, 285202 (2010)
Dain S.: Proof of the angular momentum-mass inequality for axisymmetric black hole. J. Differential Geom. 79, 33–67 (2008)
Dain S.: A variational principle for stationary, axisymmetric solutions of Einstein’s equations. Class. Quant. Grav. 23, 6857–6871 (2006)
Evans L.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)
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)
Weinstein G.: N-black hole stationary and axially symmetric solutions of the Einstein/Maxwell equations. Commun. Part. Diff. Eqs. 21, 1389–1430 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Piotr T. Chrusciel.
R. Schoen was partially supported by NSF grant DMS-1105323.
Rights and permissions
About this article
Cite this article
Schoen, R., Zhou, X. Convexity of Reduced Energy and Mass Angular Momentum Inequalities. Ann. Henri Poincaré 14, 1747–1773 (2013). https://doi.org/10.1007/s00023-013-0240-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0240-1