Volume Above Distance Below with Boundary II

TL;DR

This paper improves understanding of volume preserving intrinsic flat convergence for manifolds with boundary, showing that a boundary area bound suffices, and providing an example illustrating the necessity of such a bound.

Contribution

It demonstrates that only a boundary area bound is needed for convergence, refining previous conditions that were more restrictive.

Findings

A boundary area bound ensures volume preserving intrinsic flat convergence.

An example shows convergence may fail without a boundary area bound.

The paper extends previous results to manifolds with boundary using new estimates.

Abstract

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.