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, which completely solves an open problem due to Gromov (see Question ref{extension1}). Then we introduce a fill-in invariant (see Definition ref{fillininvariant}) 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 AH manifolds implies that for AF manifolds. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromovs conjectures formulated in cite{Gro19} (see Conjecture ref{conj0} and Conjecture ref{conj1} below)