On the moduli space of asymptotically flat manifolds with boundary and the constraint equations

Carlotto-Li have generalized Marques' path connectedness result for positive scalar curvature $R>0$ metrics on closed $3$-manifolds to the case of compact $3$-manifolds with $R>0$ and mean convex boundary $H>0$. Using their result, we show that the space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on $\mathbb{R}^{3}\backslash B^{3}$ is path connected. The argument bypasses Cerf's theorem, which was used in Marques' proof but which becomes inapplicable in the presence of a boundary. We also show path connectedness for a class of maximal initial data sets with marginally outer trapped boundary.