Summary: ON THE UNIQUENESS AND GLOBAL DYNAMICS OF ADS SPACETIMES
MICHAEL T. ANDERSON
Abstract. We study global aspects of complete, nonsingular asymptotically locally AdS space
times solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally sta
tionary spacetime. It is proved that any such solution which is asymptotically stationary to the
past and future is itself globally stationary.
This gives certain rigidity or uniqueness results for exact AdS and related spacetimes.
Consider geodesically complete, asymptotically simple solutions of the vacuum Einstein equations
with negative cosmological constant # < 0 in (n+1) dimensions. Up to rescaling, these are given by
complete, (nonsingular), metrics g, defined on manifolds of the form M n+1 = R×#, and satisfying
the Einstein equations
(1.1) Ric g = -ng.
The metric g has a conformal completion, at least C 2 , with conformal boundary (I, [#]), where #
is a complete Lorentz metric on I. Topologically, conformal infinity I is of the form R × ##.
The canonical example is the (exact) antide Sitter spacetime g AdS , which may be represented
globally in static form as
(1.2) g AdS = - cosh 2 r dt 2 + dr 2 + sinh 2 r g S n-1 (1) ,
where g S n-1 (1) is the round metric of radius 1 on the sphere S n-1 . Here M = R×R n , with conformal
infinity I = R × S n-1 , with boundary metric # 0 = -dt 2 + g S n-1 (1) the Einstein static cylinder.