Vacuum initial data on $\mathbb{S}^3$ from Killing vectors

Abstract

We construct compact initial data of constant mean curvature $\widetilde{K}$ for Einstein's 4d vacuum equations with $\widehat{\Lambda} = \Lambda - (\widetilde{K}^2/3)$ positive, where $\Lambda$ is the cosmological constant, via the conformal method. To construct a transverse, trace-free (TT) momentum tensor explicitly we first observe that, if the seed manifold has two orthogonal Killing vectors, their symmetrized tensor product is a natural TT candidate. Without the orthogonality requirement, but on locally conformally flat seed manifolds there is a generalized construction for the momentum which also involves the derivatives of the Killing fields found in work by Beig and Krammer [2]. We consider in particular the round three sphere and classify the TT tensors resulting from all possible pairs of its six Killing vectors, focusing on the commuting case where the seed data are $\mathbb{U}(1) \times \mathbb{U}(1)$ -symmetric. As to solving the Lichnerowicz equation, we discuss in particular potential "symmetry breaking" by which we mean that solutions have less symmetries than the equation itself; we compare with the case of the "round donut" of topology $\mathbb{S}^2 \times \mathbb{S}$. In the absence of symmetry breaking, the Lichnerowicz equation for a $\mathbb{U}(1) \times \mathbb{U}(1)$ symmetric momentum on $\mathbb{S}^3$ reduces to an ODE. We analyze distinguished families of solutions and the resulting data via a combination of analytical and numerical techniques. Finally we investigate marginally trapped surfaces of toroidal topology in our data.