# Alexandrov Spaces with Integral Current Structure

## TL;DR

This paper establishes that closed, orientable Alexandrov spaces can be equipped with an integral current structure, enabling analysis of their limits under Gromov-Hausdorff and intrinsic flat convergence.

## Contribution

It proves that such Alexandrov spaces admit an integral current structure with no boundary, linking geometric and measure-theoretic frameworks.

## Key findings

- Alexandrov spaces can be endowed with integral current structures
- Limits of non-collapsing sequences with curvature and diameter bounds coincide in Gromov-Hausdorff and intrinsic flat sense
- The result bridges geometric analysis and metric measure theory for Alexandrov spaces

## Abstract

We endow each closed, orientable Alexandrov space $(X, d)$ with an integral current $T$ of weight equal to 1, $\partial T = 0 and \set(T) = X$, in other words, we prove that $(X, d, T)$ is an integral current space with no boundary. Combining this result with a result of Li and Perales, we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat limits agree.