On the Gromov-Hausdorff Limits of Compact Surfaces with Boundary

TL;DR

This paper characterizes the Gromov-Hausdorff limits of compact surfaces with boundary, focusing on surfaces with fixed Euler characteristic, and extends previous results from closed to bounded surfaces.

Contribution

It provides a complete description of the limit spaces for surfaces with boundary, expanding the understanding of Gromov-Hausdorff convergence beyond closed surfaces.

Findings

Describes the topology of limit spaces with boundary

Extends previous results from closed to bounded surfaces

Analyzes the properties of the limit metric spaces

Abstract

In this work we investigate Gromov-Hausdorff limits of compact surfaces carrying length metrics. More precisely, we consider the case where all surfaces have the same Euler characteristic. We give a complete description of the limit spaces and study their topological properties. Our investigation builds on the results of a previous work which treats the case of closed surfaces.