Rotating Bowen-York initial data with a positive cosmological constant

TL;DR

This paper extends Bowen-York initial data to include a positive cosmological constant, numerically analyzing solution branches and bifurcations on a compactified domain, revealing new solution structures and their bifurcation points.

Contribution

It generalizes Bowen-York initial data to positive cosmological constant cases and identifies new solution branches and bifurcation points through numerical and bifurcation analysis.

Findings

Existence of multiple solution branches depending on variables

Identification of bifurcation points using bifurcation theory

Numerical evidence for absence of additional solution branches

Abstract

A generalization of the Bowen-York initial data to the case with a positive cosmological constant is investigated. We follow the construction presented recently by Bizo\'n, Pletka and Simon, and solve numerically the Lichnerowicz equation on a compactified domain $\mathbb S^1 \times \mathbb S^2$. In addition to two branches of solutions depending on the polar variable on $\mathbb S^2$ that were already known, we find branches of solutions depending on two variables: the polar variable on $\mathbb S^2$ and the coordinate on $\mathbb S^1$. Using Vanderbauwhede's results concerning bifurcations from symmetric solutions, we show the existence of the corresponding bifurcation points. By linearizing the Lichnerowicz equation and solving the resulting eigenvalue problem, we collect numerical evidence suggesting the absence of additional branches of solutions.