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.
Best Answer
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 $\ll \log(g)$.