Convergence of Riemannian 2-manifolds under a uniform curvature and contractibility bound
Tobias Dott
TL;DR
The paper characterizes the Gromov-Hausdorff limits of uniformly semi-locally 1-connected sequences of closed, connected Riemannian 2-manifolds all homeomorphic to a fixed surface $S$ with uniformly bounded total absolute curvature $C$. It develops a curvature-free, local-to-global framework that uses local uniform approximations and Whyburn’s cyclic-element theory to describe limit spaces: maximal cyclic subsets become surfaces with bounded curvature, the global curvature content is bounded by $C$, and singularities satisfy precise curvature inequalities involving the index and boundary rotations. A central result provides a sharp inequality: $$\sum_{n=1}^{\infty} |\omega_{T_n}|(T_n\setminus Cut_X) + \sum_{p\in Sing_X} 2\pi\,|ind_X(p)-2| + \sum_{p\in Cut_X} \sum_{n=1}^{\infty} \mathbb{1}_{T_n}(p)\,\theta_{T_n}(p) \le C,$$ linking the limit geometry to the total curvature bound. The paper also proves a converse: any limit space in the target class $\mathcal{L}(S,C)$ arises as such a GH limit, establishing a full classification of these degenerations with explicit topological and curvature data, and thus extending prior work by Burago–Shioya and relating curvature-free limit descriptions to uniform approximations.
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.