Convergence of Riemannian 2-manifolds under a uniform curvature and contractibility bound

TL;DR

This paper investigates the limits of sequences of closed Riemannian 2-manifolds with bounded curvature and contractibility, extending previous work to describe their Gromov-Hausdorff limits.

Contribution

It provides a detailed description of the Gromov-Hausdorff limits for sequences of Riemannian 2-manifolds under uniform curvature and contractibility bounds, extending prior results.

Findings

Gromov-Hausdorff limits are characterized for the given class of manifolds.

The work generalizes earlier results by Burago and Shioya.

Limits preserve certain topological and geometric properties.

Abstract

We consider uniformly semi-locally 1-connected sequences of closed connected Riemannian 2-manifolds. In particular, we assume that the manifolds are homeomorphic to each other and that their total absolute curvature is uniformly bounded. The purpose of this paper is a description of the Gromov-Hausdorff limits of such sequences. Our work extends earlier investigations by Burago and Shioya.