Anisotropic Einstein data with isotropic nonnegative scalar curvature

TL;DR

This paper constructs black hole initial data with prescribed scalar curvature, revealing anisotropic blow-up profiles through bifurcation analysis, despite isotropic curvature conditions.

Contribution

It introduces a novel method to generate anisotropic blow-up solutions in Einstein initial data with isotropic scalar curvature using bifurcation theory.

Findings

Existence of non-symmetric blow-up profiles

Application of equivariant bifurcation theory to Einstein constraints

Construction of regular metrics up to the blow-up surface

Abstract

We construct a black hole initial data for the Einstein equations with prescribed scalar curvature, or more precisely a piece of initial data contained inside the black hole. The constraints translate into a parabolic equation, with radius as "time" variable, on a metric component u that undergoes blow up. The metric itself is regular up to and including the surface at blow up radius, which is a minimal surface. By applying equivariant bifurcation theory on a self similarly rescaled equation, we show the existence of blow up profiles that are not O(3) symmetric - or anisotropic - although the curvature was isotropically prescribed.