The pointed intrinsic flat distance between locally integral current spaces

TL;DR

This paper introduces a new distance measure for pointed locally integral current spaces and proves its equivalence to the Lang-Wenger convergence, facilitating a compactness theorem expressed via this distance.

Contribution

It defines a pointed intrinsic flat distance for locally integral current spaces and establishes its equivalence with Lang-Wenger convergence, enabling a metric-based compactness theorem.

Findings

The new distance characterizes convergence of locally integral current spaces.

Convergence in this distance is equivalent to Lang-Wenger convergence.

The work provides a metric formulation of the Lang-Wenger compactness theorem.

Abstract

In this note we define a distance between two pointed locally integral current spaces. We prove that a sequence of pointed locally integral current spaces converges with respect to this distance if and only if it converges in the sense of Lang-Wenger. This enables us to state the compactness theorem by Lang-Wenger for pointed locally integral current spaces in terms of a distance function.