Remarks on mass and angular momenta for U(1)²-invariant initial data

We extend Brill's positive mass theorem to a large class of asymptotically flat, maximal, U(1)²-invariant initial data sets on simply connected four dimensional manifolds Σ. Moreover, we extend the local mass angular momenta inequality result [A. Alaee and H. K. Kunduri, Classical Quantum Gravity 32(16), 165020 (2015)] for U(1)² invariant black holes to the case with nonzero stress energy tensor with positive matter density and energy-momentum current invariant under the above symmetries.