A Precompactness theorem for Einstein metrics on homogeneous torus bundles

We investigate Einstein metrics on the total space of homogeneous torus bundles.

Allowing for different embeddings of the isotropy group gives rise to infinite families of simply connected homogeneous spaces fibred over a common simply connected base, with toral fibres of fixed dimension.

Descending to the common base, we establish a set of estimates that we use to prove a precompactness theorem for Einstein manifolds in these families.