  It was shown by B. Allen, R. Perales, and C. Sormani that on a closed
manifold where the diameter of a sequence of Riemannian metrics is bounded, if
the volume converges to the volume of a limit manifold, and the sequence of
Riemannian metrics are $C^0$ converging from below then one can conclude volume
preserving Sormani-Wenger Intrinsic Flat convergence. The result was extended
to manifolds with boundary by B. Allen and R. Perales by a doubling with necks
procedure which produced a closed manifold and reduced the case with boundary
to the case without boundary. The consequence of the doubling with necks
procedure was requiring a stronger condition than necessary on the boundary.
Using the estimates for the Sormani-Wenger Intrinsic Flat distance on manifolds
with boundary developed by B. Allen and R. Perales, we show that only a bound
on the area of the boundary is needed in order to conclude volume preserving
intrinsic flat convergence for manifolds with boundary. We also provide an
example which shows that one should not expect convergence without a bound on
area.
