Abstract:

Motivated by the conformal method of solving the constraints in General Relativity,

R. Beig and W. Krammer constructed, on any 3‐dimensional conformally flat Riemannian

manifold M, a symmetric, tracefree two‐tensor out of an arbitrary vector V and a

conformal Killing vector W (and their derivatives). If V is divergence free, so is the Beig‐

Krammer tensor ‐ hence it can serve as ADM momentum density in vacuum, (possibly

with cosmological constant). We examine the very special case that M is the round three

sphere and that V and W are Killing vectors, and compare with the known "donut" case.

The application of this tensor to the initial value problem becomes particularly

interesting in view of a recent theorem by Premoselli which in essence settles the

question of (non‐)existence of solutions of the Lichnerowicz equation on compact

Riemannian three manifolds. This is joint and ongoing work with Piotr Bizon.