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. 
P. Bizo\'n and A. Wasserman