Existence of black hole solutions for the Einstein-Yang/Mills equations (Q2367966)

The paper provides the proof of the existence of an infinite number of black holes solutions for any event horizon (for a radius \(r = r_ H\)) with gauge group \(SU(2)\). Each solution corresponds to a finite Arnowitt- Deser-Miser (ADM) mass. The associated Einstein metric tends to the Schwarzschild metric at a rate \(1/r^ 2\) as \(r\) tends to infinity. Writing the metric as \(ds^ 2 = -(dt/T(r))^ 2 + dr^ 2/A(r) + (\text{rd }\Omega)^ 2\), the unknown functions (defining orbits) are \(T\), \(A\) together with the function \(w\) which occurs in the curvature 2- form \(F = w'\tau_ 1 dr \wedge d\vartheta + w'\tau_ 2 dr \wedge (\sin \vartheta d\varphi) - (1-w^ 2)\tau_ 3 d\vartheta\wedge(\sin \vartheta d\varphi)\). The EYM equations are formulated in these functions and in order to obtain black hole solutions are required the initial data \(A = 0\), \(| w| < 1\), \(| w'| < \infty\) for finite functions \(A\), \(w\), \(w'\) of \(r\). For \(w'\) bounded near \(r_ H\), the limit of \(w'\) exists at \(r_ H\) and \((w,w')\) at \(r_ H\) lie on a curve \(\mathcal C\) in the \(w\), \(w'\) plane. There exists a sequence of initial values on \(\mathcal C\) for which the solutions are not singular \((A>0)\) and bounded if \(r > r_ H\). Each solution has a finite ADM mass: \(\lim r(1-A)\) for \(r\to \infty\) which can be arbitrarily large. The quite ample proofs are based on a main theorem and a set of lemmas and propositions which refer to different curves \(\mathcal C\) of continuous type or with discontinuity from which follows that the orbits crush or not.