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Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that there are $\epsilon$-Gromov-Hausdorff approximations between $(M,g)$ and $(N,h)$.

Then define $d(M,N)$ to be the infimum of the Gromov-Hausdorff distance between $(M,g)$ and $(N,h)$ taken over all Riemannian metrics $g$ and $h$ with sectional curvatures bounded in absolute value by 1.

If you know $d(M,N)$ for various choices of $N$, what can you conclude about $M$?

I know that Cheeger-Fukaya-Gromov theory on collapsed manifolds exactly covers the case when $d(M,N)=0$; for example $d(M,pt)=0$ if and only if $M$ is almost flat. I'm interested in situations where these numbers do not vanish. For example:

If $\Sigma$ is some compact orientable surface how should $d(\Sigma,pt)$ depend on the genus?

If $S^n$ is the standard sphere of dimension $n$, to what extent do the numbers $d(M,S^n)$ determine $M$? Are there manifolds $M$ and $N$ for which $d(M,N)\neq 0$ but $d(M,S^n)=d(N,S^n)$ for all $n$?

Is there an $\epsilon$ depending only on dimensions such that $d(M,N)<\epsilon$ implies $d(M,N)=0$?

If anyone can point me toward a reference/paper it would be very appreciated.

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  • $\begingroup$ If the Cheeger-Fukaya-Gromov theory gives that alost-flat result, in what way does it fail to give effective lower bounds in the cases you're looking at? $\endgroup$ Mar 28, 2016 at 21:11
  • $\begingroup$ I think the quick response to this comment is: my questions enter into "What exactly is the GH distance between two spaces?" territory, for which the correct answer is almost always "Who cares unless it can be made small." I can't see how any of the classical collapsing theory applies to the $d(M,N)>0$ cases, and I feel that your question is exactly related to what I'm asking. If someone can see a way to get effective bounds, I hope they post an answer. $\endgroup$ Mar 28, 2016 at 21:57
  • $\begingroup$ Yes, I see how what I'm asking looks like a paraphrase of the question, but I was really thinking of: how did they prove the "only if" part? $\endgroup$ Mar 28, 2016 at 22:35
  • $\begingroup$ $d(M,pt)=0$ is a rephrasing of the definition of being almost flat once you remark that the Gromov-Hausdorff distance to a point is proportional to the diameter of the space. There is not much to prove when it comes to obtaining "$d(M,pt)=0$ if and only if $M$ is almost flat", just unwind the definitions. Is this the only if you're talking about? $\endgroup$ Mar 28, 2016 at 22:52
  • $\begingroup$ Ah! Hmm... whereas in the case you're considering... e.g. specializing to even-dimensional manifolds perhaps with interesting Euler characteristic, asking for small curvature means the volume has to be large, because $\int Pf dVol$ is fixed... $\endgroup$ Mar 30, 2016 at 4:14

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This is only an answer to one point of your question: for surfaces of large genus $g$ the distance should be $$ d(S, \mathrm{point}) \asymp \log(g). $$ The lower bound should follow from volume estimates (by Gauss-Bonnet the volume is at least $\gg g$ and the volume growth of balls in spaces with bounded curvature is at most exponential).

The upper bound can be proven at least in two ways, explicit construction and random. First it follows from the existence of expanding sequences of covers of arithmetic surfaces, which have diameter $\ll \log(g)$ by general facts about expanders (see eg. Lubotzky's book Discrete groups, expanding graphs and invariant measures, in particular 7.3.11(ii)). I think that probably one can get any genus this way but I am unsure whether somebody wrote it down. On the other hand, by results of Mirzakhani (see II) in the intro of Growth of Weil--Petersson volumes and random hyperbolic surfaces of large genus, 1012.2167) a typical surface of genus $g$ has diameter $\ll \log(g)$.

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  • $\begingroup$ I knew of Mirzakhani's results on the diameter but missed the lower bound. Thank you! $\endgroup$ Mar 31, 2016 at 16:08

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