Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, completely solving an open problem due to Gromov (see Question 1.1). Then we introduce a fill-in invariant (see Definition 1.2) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for All manifolds implies that for AF manifolds via this fill-in invariant. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromov's two conjectures formulated in [M. Gromov, Four lectures on scalar curvature, preprint 2019] (see Conjecture 1.1 and Conjecture 1.2 below).