New invariants of Gromov-Hausdorff limits of Riemannian surfaces with curvature bounded below

TL;DR

This paper introduces new invariants for Gromov-Hausdorff limits of Riemannian surfaces with curvature bounds, providing a classification and existence results for these invariants in the case of 2-surfaces.

Contribution

It proves the existence of a geometric invariant function for limits of 2-surfaces and classifies all possible such functions, extending previous unproven ideas.

Findings

Existence of the invariant function for sequences of 2-surfaces.

Classification of possible invariant functions.

Connection to intrinsic volumes of the surfaces.

Abstract

Let $\{X_i\}$ be a sequence of compact $n$-dimensional Alexandrov spaces (e.g. Riemannian manifolds) with curvature uniformly bounded below which converges in the Gromov-Hausdorff sense to a compact Alexandrov space $X$. In an earlier paper by the first author there was described (without a proof) a construction of an integer valued function on $X$; this function carries additional geometric information on the sequence such as the limit of intrinsic volumes of $X_i$'s. In this paper we consider sequences of closed 2-surfaces and (1) prove the existence of such a function in this situation; and (2) classify the functions which may arise from the construction.