  Given a constant C and a smooth closed $(n-1)$-dimensional Riemannian
manifold $(\Sigma, g)$ equipped with a positive function $H$, a natural
question to ask is whether this manifold can be realised as the boundary of a
smooth $n$-dimensional Riemannian manifold with scalar curvature bounded below
by C. That is, does there exist a fill-in of $(\Sigma,g,H)$ with scalar
curvature bounded below by C?
  We use variations of an argument due to Miao and the author
[arXiv:1701.04805] to explicitly construct fill-ins with different scalar
curvature lower bounds, where we permit the fill-in to contain another boundary
component provided it is a minimal surface. Our main focus is to illustrate the
applications of such fill-ins to geometric inequalities in the context of
general relativity. By filling in a manifold beyond a boundary, one is able to
obtain lower bounds on the mass in terms of the boundary geometry through
positive mass theorems and Penrose inequalities. We consider fill-ins with both
positive and negative scalar curvature lower bounds, which from the perspective
of general relativity corresponds to the sign of the cosmological constant, as
well as a fill-in suitable for the inclusion of electric charge.
