On periodic solutions of nonlinear wave equations, including Einstein equations with a negative cosmological constant

We construct periodic solutions of nonlinear wave equations using analytic continuation. The construction applies in particular to Einstein equations, leading to infinite-dimensional families of time-periodic solutions of the vacuum, or of the Einstein-Maxwell-dilaton-scalar fields-Yang-Mills-Higgs-Chern-Simons-$f(R)$ equations, with a negative cosmological constant.Comment: 15 pages, 1 figur

On rigidity of analytic black holes

We establish global extendibility (to the domain of outer communications) of locally defined isometries of appropriately regular analytic black holes. This allows us to fill a gap in the Hawking-Ellis proof of black-hole rigidity.Comment: 10 pages, latex with amssym

Bifurcating solutions of the Lichnerowicz equation

We give an exhaustive description of bifurcations and of the number of solutions of the vacuum Lichnerowicz equation with positive cosmological constant on $S^1\times S^2$ with $U(1)\times SO(3)$-invariant seed data. The resulting CMC slicings of Schwarzschild-de Sitter and Nariai are described.Comment: 33 pages, 25 figure

On higher dimensional black holes with abelian isometry group

We consider (n+1)--dimensional, stationary, asymptotically flat, or Kaluza-Klein asymptotically flat black holes, with an abelian $s$--dimensional subgroup of the isometry group satisfying an orthogonal integrability condition. Under suitable regularity conditions we prove that the area of the group orbits is positive on the domain of outer communications, vanishing only on its boundary and on the "symmetry axis". We further show that the orbits of the connected component of the isometry group are timelike throughout the domain of outer communications. Those results provide a starting point for the classification of such black holes. Finally, we show non-existence of zeros of static Killing vectors on degenerate Killing horizons, as needed for the generalisation of the static no-hair theorem to higher dimensions

Stationary Black Holes

We review the theory of stationary black hole solutions of vacuum Einstein equations.Comment: 10 pages, 2 figures, to appear in Encyclopedia of Mathematical Physics in 200

Mass and angular-momentum inequalities for axi-symmetric initial data sets I. Positivity of mass

We extend the validity of Brill's axisymmetric positive energy theorem to all asymptotically flat initial data sets with positive scalar curvature on simply connected manifolds.Comment: 33 pages in A