Gromov-Hausdorff Convergence of Metric Quotients and Singular Conic-Flat Surfaces

TL;DR

This paper establishes conditions under which metric quotients of a space converge in the Gromov-Hausdorff sense, with applications to sequences of conic-flat surfaces exhibiting singularities and unbounded curvature.

Contribution

It provides new sufficient conditions for Gromov-Hausdorff convergence of metric quotients, especially in the context of conic-flat surfaces with singularities.

Findings

Convergence criteria for metric quotients are identified.

Examples of conic-flat spheres converging to singular conic-flat spheres are constructed.

Some limit surfaces have unbounded curvature at singularities.

Abstract

Given metric quotients $S$ and $S_n$, $n \in \mathbb{N}$, of a metric space $X$, sufficient conditions are provided on the data defining them guaranteeing that $S$ is the Gromov-Hausdorff limit of $S_n$. These conditions are recognized within metric quotients of plane polygons determined by side-pairings known as plain paper-folding schemes. In particular, concrete examples are given of sequences of two-dimensional conic-flat spheres converging to spheres that are conic-flat except around certain singularities, some of them with unbounded curvature in the sense of comparative geometry.