Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere

We prove existence of a countable family of spherically symmetric self-similar wave maps from 3+1 Minkowski spacetime into the 3-sphere. These maps can be viewed as excitations of the ground state wave map found previously by Shatah. The first excitation is particularly interesting in the context of the Cauchy problem since it plays the role of a critical solution sitting at the threshold of singularity formation. We analyze the linear stability of our wave maps and show that the number of unstable modes about a given map is equal to its excitation index. Finally, we formulate a condition under which these results can be generalized to higher dimensions.Comment: 15 pages, extended version to appear in Comm. Math. Phy

Globally regular instability of AdS3AdS_3

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 $AdS_3$ 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 $AdS_3$. 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 $\rho(t)$ of the analyticity strip of solutions in the complex spatial plane and argue by extrapolation that $\rho(t)$ 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.Comment: 5 pages, 4 figures. Dedicated to Professor Andrzej Staruszkiewicz on the occasion of the 50th anniversary of his pioneering work on three-dimensional gravit

A note on the non-existence of σ\sigma-model solitons in the 2+1 dimensional AdS gravity

We show that the gravitating static soliton in the 2+1 dimensional O(3) $\sigma$ model does not exist in the presence of a negative cosmological constant

Self-similar spherically symmetric wave maps coupled to gravity

We investigate spherically symmetric continuously self-similar (CSS) solutions in the SU(2) sigma model coupled to gravity. Using mixed numerical and analytical methods, we provide evidence for the existence (for small coupling) of a countable family of regular CSS solutions. This fact is argued to have important implications for the ongoing studies of black hole formation in the model.Comment: 13 pages, 8 figure