Gromov-Hausdorff convergence of metric pairs and metric tuples

TL;DR

This paper explores the Gromov-Hausdorff convergence for metric pairs and tuples, establishing foundational theorems and extending classical results to more complex metric structures.

Contribution

It introduces new definitions, proves their equivalence, and extends key theorems like embedding, completeness, and compactness to metric pairs and tuples.

Findings

Proved equivalence of different Gromov-Hausdorff convergence definitions.

Established embedding, completeness, and compactness theorems for metric pairs and tuples.

Extended Fukaya's and Grove-Petersen--Wu's theorems to stratified spaces.

Abstract

We study the Gromov-Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov--Hausdorff equivariant convergence and a version of Grove-Petersen--Wu's finiteness theorem for stratified spaces.