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

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 [1] 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.