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

2022/04/27 by Semyon Alesker, Alesker, Semyon, Mikhail G. Katz +3
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2204.13018

openalex publication_date 2022/04/27 · openalex created_date 2022/05/01 · openalex updated_date 2026/07/28

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.

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