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Given two metric spaces we can define the Gromov-Hausdorff (GH) distance. There are compactness results stating that a sequence of manifolds of a fixed dimension, with a uniform lower Ricci bound and a uniform diameter upper bound (not necessary if we work we pointed-GH) GH-converges, up to a subsequence, to a metric space (see Petersen Theorem pag 300).

My question is what happens if we do not assume the uniform dimensional bound. In this case it is harder to recover Gromov's covering criterion (Petersen pag 299), but are there reasonable assumptions to have precompactness? Or, is there another notion of convergence that allows us to pass to infinite dimension?

For example, what can we say about the sequence of unit spheres $\mathcal{S}^n$?

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Theorem. For all integers $0<m<n$ $d_{GH}(\mathbb{S}^m,\mathbb{S}^n)\geq \frac{\pi}{4}$.

This is Theorem B on page 7 in

S. Lim, F. Mémoli, Z. Smith, The Gromov-Hausdorff distance between spheres. arXiv:2105.00611.

Therefore no sequence of spheres of different dimensions can converge in the Gromov-Hausdorff metric.

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Gromov has such a general result in

Gromov, Misha. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Reprint of the 2001 English edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2007. xx+585 pp. ISBN: 978-0-8176-4582-3; 0-8176-4582-9

On page 273 you will find Proposition 5.2 asserting existence of a limit metric space.

This is followed on page 275 by a more specific result for sequences of manifolds of a fixed dimension, relying on Bishop's inequality.

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  • $\begingroup$ This is the same result of Petersen at page 299, which also says that the condition is necessary for precompactness. I think that $\mathcal{S}^n$ does not satisfy such condition though, and hence I was wondering if there is another notion of convergence that allows to converge to an infinite dimensional space. $\endgroup$
    – theflame
    Commented Nov 23, 2023 at 14:29

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