Differentiable manifolds whose Ricci curvature is proportional
to the metric are called Einstein manifolds. Such manifolds have been
central objects of study in differential geometry and Einstein's theory
for general relativity, with some strong recent results. In this talk, we
shall focus on positively curved 3+1 Lorentzian Einstein manifolds with
one spacelike rotational isometry. After performing the dimensional
reduction to a 2+1 dimensional Einstein's equations coupled to 'shifted'
wave maps, we shall prove two explicit positive mass theorems:

i)      for CMC slices in expanding region
ii)     for maximal slices in 'stationary' region.