  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 Gromov's conjectures formulated in \cite{Gro19} (see Conjecture
\ref{conj0} and Conjecture \ref{conj1} below)
