Properties of the Intrinsic Flat Distance

Abstract

Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorff and intrinsic flat compactness theorems including the Tetrahedral Compactness Theorem. Much of the work in this paper builds upon Ambrosio-Kirchheim's Slicing Theorem combined with an adapted version Gromov's Filling Volume.

V1-V2: by Sormani included F to GH Conv Thm, Arz-Asc Thms and BW Thms that were then simplified and moved to arXiv:1402.6066 and also an incomplete proof of the Tetrahedral Compactness Thm. V3: has a new coauthor Portegies and new Section 3. V4: new sections 4 and 5. V5: added new Appendix