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Consider a closed 2-dimensional surface (not necessarily orientable) with a metric with curvature at least -1 in the sense of Alexandrov. It it true that on this surface there is a sequence of infinitely smooth riemannian metrics of Gauss curvature say at least -100 which converges to the given metric in the Gromov-Hausdorff sense?

A reference would be helpful.

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    $\begingroup$ With $-110$ replaced by some negative constant this goes back to Alexandrov-Zalgaller and Reshetnyak. I would read [Yu. G. Reshetnyak, "Two-dimensional manifolds of bounded curvature”, MR1263963 (94i:53038)]. $\endgroup$
    – Igor Belegradek
    Commented Aug 12, 2019 at 14:04
  • $\begingroup$ @IgorBelegradek : Thanks. This certainly answers my question. Would you like to write it as an answer? $\endgroup$
    – asv
    Commented Aug 12, 2019 at 14:25
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    $\begingroup$ It seems one can approximate by Riemannian metrics with the same lower curvature bound. This is what is claimed in Theorem A.1 of arxiv.org/abs/1204.5461 which also contains a nice summary of basic facts about Alexandrov surfaces. I don't have a firm grip on this material and hence would rather not post my comments as an answer. I encourage you to check everything and then post it as an answer. $\endgroup$
    – Igor Belegradek
    Commented Aug 12, 2019 at 14:39
  • $\begingroup$ @IgorBelegradek : My impression is that Reshetnyak’s paper you mentioned discusses spaces of bounded curvature in a different sense: not bounded below (as I asked) but bounded in some integral sense. Theory of such spaces is also developed in Alexandrov-Zalgaller book. Unfortunately I did not find the statement relevant in my case. However it is stated explicitly in the paper by Richard you mentioned without any proof or reference. $\endgroup$
    – asv
    Commented Aug 13, 2019 at 11:45
  • $\begingroup$ Any (possibly non-smooth) surface of curvature bounded below has bounded integral curvature, and this is explaned in Reshetnyak's article (which is detailed but not written elegantly, in my opinion). The approximation by Riemannian metrics holds in this larger class. Let me give a few more references, section 3 in hal.archives-ouvertes.fr/hal-01967250/document, and arxiv.org/abs/0906.3407 or crm.sns.it/media/course/2102/SlidesPisa.pdf where you can learn more about bounded integral curvature on non-smooth surfaces. Also see arxiv.org/abs/1605.07755. $\endgroup$
    – Igor Belegradek
    Commented Aug 13, 2019 at 12:14

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