Communications in Analysis and Geometry

Volume 29 (2021)

Number 1

Alexandrov spaces with integral current structure

Pages: 115 – 149



M. Jaramillo (Mathematics Q Center Hartford, University of Connecticut, Hartford, Ct., U.S.A.)

R. Perales (Conacyt Research Fellow. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Oaxaca, México)

P. Rajan (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.)

C. Searle (Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, Kansas, U.S.A.)

A. Siffert (Westfälische Wilhelms-Universität Münster, Germany)


We endow each closed, orientable Alexandrov space $(X, d)$ with an integral current $T$ of weight equal to $1 , \partial T = 0$ and $\operatorname{set}(T) = X$, 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.

Received 28 March 2017

Accepted 31 March 2018

Published 11 March 2021