## 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. The paper (Alesker in Arnold Math J 4(1):1–17, 2018) outlined (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 the X_{i}. 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.

Original language | English |
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Article number | 12 |

Journal | Geometriae Dedicata |

Volume | 217 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2023 |

### Bibliographical note

Publisher Copyright:© 2022, The Author(s), under exclusive licence to Springer Nature B.V.

### Funding

R. Prosanov: This research was funded in part by the Swiss National Science Foundation grant and was funded in part by the Austrian Science Fund (FWF) ESPRIT Grant ESP-12-N. We express our gratitude to V. Kapovitch who supplied the proof of Theorem 6.14. The first author is very grateful to A. Petrunin for numerous useful discussions. The third author is also grateful to his postdoctoral mentor M. Eichmair for his support. Part of this research was done while the third author was visiting the Tel Aviv University, and the third author is very grateful to the institution for its hospitality. M. G. Katz: Partially supported by the BSF Grant 2020124 and the ISF Grant 743/22 S. Alesker: Partially supported by the US-Israel BSF grant 2018115 and the ISF Grant 743/22.

Funders | Funder number |
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US-Israel BSF | 2018115 |

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung | |

United States-Israel Binational Science Foundation | 2020124 |

Austrian Science Fund | ESP-12-N |

Israel Science Foundation | 743/22 |

Tel Aviv University |

## Keywords

- Alexandrov surfaces
- Gromov-Hausdorff convergence
- Riemannian surfaces