We point out that the weakly turbulent instability of anti-de Sitter space, recently found in [1] for four dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant, is present in all dimensions d + 1 for d ≥ 3, contrary to a claim made in [2].

We consider three-dimensional AdS gravity minimally coupled to a massless scalar field and study numerically the evolution of small smooth circularly symmetric perturbations of the AdS3 spacetime. As in higher dimensions, for a large class of perturbations, we observe a turbulent cascade of energy to high frequencies which entails instability of AdS3. However, in contrast to higher dimensions, the cascade cannot be terminated by black hole formation because small perturbations have energy below the black hole threshold. This situation appears to be challenging for the cosmic censor. Analysing the energy spectrum of the cascade we determine the width ρ(t) of the analyticity strip of solutions in the complex spatial plane and argue by extrapolation that ρ(t) does not vanish in finite time. This provides evidence that the turbulence is too weak to produce a naked singularity and the solutions remain globally regular in time, in accordance with the cosmic censorship hypothesis.Dedicated to Professor Andrzej Staruszkiewicz on the occasion of the 50th anniversary of his pioneering work on three-dimensional gravity [1].

We carry out numerical experiments in the critical collapse of a spherically symmetric massless scalar field in 2+1 spacetime dimensions in the presence of a negative cosmological constant and compare them against a new theoretical model. We approximate the true critical solution as the n = 4 Garfinkle solution, matched at the lightcone to a Vaidya-like solution, and corrected to leading order for the effect of Λ < 0. This approximation is only C 3 at the lightcone and has three growing modes. We conjecture that pointwise it is a good approximation to a yet unknown true critical solution that is analytic with only one growing mode (itself approximated by the top mode of our amended Garfinkle solution). With this conjecture, we predict a Ricci-scaling exponent of γ = 8/7 and a mass-scaling exponent of δ = 16/23, compatible with our numerical experiments. CONTENTS