On the moduli space of asymptoticallyflat manifolds with boundary and theconstraint equations

Carlotto-Li have generalized Marques' path connectedness resultfor positive scalar curvatureR >0 metrics on closed 3-manifoldsto the case of compact 3-manifolds withR >0 and mean convexboundaryH >0. Using their result, we show that the space ofasymptotically flat metrics with nonnegative scalar curvature andmean convex boundary onR3\B3is path connected. The argumentbypasses Cerf's theorem, which was used in Marques' proof butwhich becomes inapplicable in the presence of a boundary. We alsoshow path connectedness for a class of maximal initial data setswith marginally outer trapped boundary