We construct time-symmetric black hole initial data for the Einstein equations with prescribed scalar curvature, or more precisely a piece of such initial data contained inside the black hole. In this case, the Einstein constraint equations translate into a parabolic equation, with radius as 'time' variable, for a metric component u that undergoes blow up. The metric itself is regular up to and including the surface at the blow up radius, which is a minimal surface. We show the existence of Einstein constrained data with blow up profiles that are anisotropic (i.e. not O(3) symmetric) although the scalar curvature was isotropically prescribed. Our results are based on center manifold theory for quasilinear parabolic equations and on equivariant bifurcation theory for not necessarily self-similar solutions of a self-similarly rescaled equation.
|Original language||English (US)|
|Number of pages||28|
|Journal||Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire|
|State||Published - Mar 1 2015|
ASJC Scopus subject areas
- Mathematical Physics
- Applied Mathematics