Construction of vacuum initial data by the conformally covariant split system

Abstract

Using the implicit function theorem, we prove existence of solutions of the so-called conformally covariant split system on compact 3-dimensional Riemannian manifolds. They give rise to non-Constant Mean Curvature (non-CMC) vacuum initial data for the Einstein equations. We investigate the conformally covariant split system defined on compact manifolds with or without boundaries. In the former case, the boundary corresponds to an apparent horizon in the constructed initial data. The case with a cosmological constant is then considered separately. Finally, to demonstrate the applicability of the conformal covariant split system in numerical studies, we provide numerical examples of solutions on manifolds $\mathbb S^1 \times \mathbb S^2$ and $\mathbb S^1 \times \mathbb T^2$.