Volume above distance below with boundary II

Abstract

It was shown by Allen (in: Volume above distance below, 2020) 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 Allen et al. (in: Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, 2021) 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 Allen et al. (in: Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, 2021), 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.