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

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A Correction to this article was published on 18 January 2023

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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.

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Notes

  1. Before Fujioka’s work [11], Petrunin [26] mentioned to the first author that he has a proof, although did not publish it.

  2. This construction was first suggested as a conjecture by the first author. Then Petrunin (see the discussion in [1]) mentioned that he has a proof. Very recently Fujioka [11] posted a proof on the arxiv.

  3. It is likely that this result is folklore.

  4. This is equation (2.5) in [34]; see references therein.

  5. The only difference between this definition and the original GH-distance is that d has to be G-invariant.

  6. Recall that a subset \(S\subset X\) is called \(\varepsilon \)-net if its \(\varepsilon \)-neighborhood equals X: \(X=S_\varepsilon \).

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Mathematics, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Semyon Alesker

  2. Department of Mathematics, Bar Ilan University, 5290002, Ramat Gan, Israel

    Mikhail G. Katz

  3. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Roman Prosanov

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  1. Semyon Alesker
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  2. Mikhail G. Katz
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Correspondence to Roman Prosanov.

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S. Alesker: Partially supported by the US-Israel BSF grant 2018115 and the ISF Grant 743/22.

M. G. Katz: Partially supported by the BSF Grant 2020124 and the ISF Grant 743/22

R. Prosanov: This research was funded in part by the Swiss National Science Foundation grant \(200021_-179133\) and was funded in part by the Austrian Science Fund (FWF) ESPRIT Grant ESP-12-N.

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Alesker, S., Katz, M.G. & Prosanov, R. New invariants of Gromov–Hausdorff limits of Riemannian surfaces with curvature bounded below. Geom Dedicata 217, 12 (2023). https://doi.org/10.1007/s10711-022-00739-x

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