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

Remark on non-existence of self-similar solutions in 4+1 vacuum gravitational collapse

We give a very short proof that the vacuum Einstein equations in 4+1 dimensions have no cohomogeneity-two Bianchi IX continuously self-similar solutions.Comment: 2 page

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

On the rank of a product of manifolds

This note gives an example of closed smooth manifolds $M$ and $N$ for which the rank of $M\times N$ is strictly greater than $rank M + rank N$

Extending algebraic actions

Sin resume

Self-similar solutions of semilinear wave equations with a focusing nonlinearity

We prove that in three space dimensions a nonlinear wave equation $u_{tt}-\Delta u = u^p$ with $p\geq 7$ being an odd integer has a countable family of regular spherically symmetric self-similar solutions.Comment: 12 pages, 3 figures, minor corrections to match the published versio

Uniqueness of zero surface gravity SU(2) Einstein–Yang/Mills black holes

In this paper we prove that the only spherically symmetric black hole solution to the SU(2) Einstein–Yang/Mills equations that has zero temperature at the event horizon is the extreme Reissner–Nordström solution. No assumptions are made on the signs of the metric coefficients, save that the metric has Lorentz signature. © 1996 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70805/2/JMAPAQ-37-3-1461-1.pd

On the existence of self-similar spherically symmetric wave maps coupled to gravity

We present a detailed analytical study of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. Using a shooting argument we prove that there is a countable family of solutions which are analytic inside the past self-similarity horizon. In addition, we show that for sufficiently small values of the coupling constant these solutions possess a regular future self-similarity horizon and thus are examples of naked singularities. One of the solutions constructed here has been recently found as the critical solution at the threshold of black hole formation.Comment: 15 pages, LaTe