The intrinsic flat distance between Riemannian manifolds and other integral current spaces

Abstract

Inspired by the Gromov-Hausdorff distance, we define a new notion called the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim's extension of Federer-Fleming's notion of integral currents to arbitrary metric spaces.We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably H m rectifiable spaces and present numerous examples. Contents

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

“…There are (at least) four different ways to introduce a convergence notion for p.m.m. spaces (for a different approach in the case of counting H N -rectifiable spaces, see [41]); here we briefly recall the main ideas; for the details the reader is referred to Subsection 3.2 and in particular to the equivalence Theorem 3.15, showing that all the approaches lead to equivalent definitions, which thereby characterize the pmG-convergence.…”

Contrasting various notions of convergence in geometric analysis

“…Intrinsic Flat Convergence. The Intrinsic Flat Distance (F) defined by Sormani-Wenger in [SW11] is defined for a large class of metric spaces called integral current spaces. In their paper they show that F is a weaker notion than Gromov Lipschitz convergence that is distinct from GH convergence and can give different limits.…”

“…It is well knows that even if one starts with a sequence of Riemannian manifolds M j = (M, g j ) that the F and GH limits need not even be Riemannian manifolds. See [Gro81b] and [SW11] for these examples. It should also be noted that GH and F limits need not agree, although if a GH limit exists and the sequence (M, g j ) has a uniform upper bound on volume, then a subsequence has a F limit and the F limit is either the zero space or a subset of the GH limit [SW11].…”

“…See [Gro81b] and [SW11] for these examples. It should also be noted that GH and F limits need not agree, although if a GH limit exists and the sequence (M, g j ) has a uniform upper bound on volume, then a subsequence has a F limit and the F limit is either the zero space or a subset of the GH limit [SW11]. There are many examples with no GH limit, that have F limits [SW11].…”

“…It should also be noted that GH and F limits need not agree, although if a GH limit exists and the sequence (M, g j ) has a uniform upper bound on volume, then a subsequence has a F limit and the F limit is either the zero space or a subset of the GH limit [SW11]. There are many examples with no GH limit, that have F limits [SW11]. There are a few theorems which provide hypotheses proving that F and GH limits exist and agree: see the work of Wenger, Matveev, Portegies, Perales, Nuñez-Zimbron, Huang, Lee, and the second author in…”