Sorry, I misuse the concept of quasi-isometry, I mean almost isometry(also called a Hausdorff approximation).


As we known, isometric Riemann manifolds have the same spectrum of Laplace-Beltrami. And it defined a class of isospectral manifolds which is a highly identical signature of manifold. However, in application, almost-isometry is more useful. Does anyone provide me an overview or reference of the relationship between spectrum and almost-isometry?


Almost isometry say for two metric space(Riemann manifold). there exist $\varepsilon$ and $f: X\rightarrow Y$ s.t.


1. $|d(x,y)-d(f(x),f(y))|<\varepsilon$ for $x,y\in X$
2. for any point $y\in Y$, there exists an $x\in X$ s.t. $d(f(x),y)<\varepsilon$


the question is:
Given two riemann manifold, how to check almost isometry and estimate inf $ \{\varepsilon\}$ from spectrum data.



Relationship between spectrum geometry and almost-isometry
The above comments are still mostly valid with $\epsilon$-isometries:


if $(M,g)$ and $(N,g)$ are Riemannian manifolds with diameters less than $D$, then


$f:M\to N$, $x\mapsto n\_0$ for some $n\_0\in N$ has


$$
|d\_N(f(x),f(x')) - d\_M(x,x')| = d\_M(x,x') \leq D
$$


and for all $y \in N$, because
$$
d\_N(y,n\_0) \leq D
$$


this $f$ is a $D$-isometry. So, just having a $\epsilon$-isometry for some large $\epsilon$ should not be seen as an unlikely occurrence (however, the (even weaker) notion of quasi-isometry that you used before *is* an interesting idea, but some sort noncompactness plays a big role in things, as should be obvious from all of these answers.




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On the other hand, $\epsilon$-isometries and the spectrum are certainly related in some ways, particularly with lower Ricci bounds:


For example in the paper by Cheeger and Colding, "On the structure of spaces with Ricci curvature bounded below. III." (J. Differential Geom., 54(1):37–74, 2000.) they prove that for manifolds with appropriate Ricci lower bounds, under Gromov-Hausdorff convergence the spectrum and eigenfunctions converge in some sense. 


For example, their Theorem 7.11 says:



> 
> 
> > 
> > For $M\_1^n$, $M\_2^n$, Riemannian manifolds satisfying
> >  $$ Ric \geq -(n-1) $$
> >  and 
> >  $$ diam(N\_1^n) \leq d < \infty $$
> >  Then for all $N < \infty$ and $\epsilon > 0$ there is a $\delta(n,d,\epsilon,N) > 0$ such that if 
> >  $$
> > d\_{GH}(M\_1^n,M\_2^n) < \delta
> > $$
> >  then for $j\leq N$, we have that $|\lambda\_{j,1} - \lambda\_{j,2}| <\epsilon$, where $\lambda\_{i,k}$ is the $i$-th eigenvalue on the $M\_k$.
> > 
> > 
> > 
> 
> 
> 


You can get more information about this [here](http://arxiv.org/abs/math/0612107), which also links to some interesting looking work by Lott about how the same question with the laplacian on $p$-forms.




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Just to clarify how $d\_{GH}$ is related to $\epsilon$-isometries, in case it is unclear. It is a theorem that the Gromov-Hausdorff distance is $<\epsilon$ if there exists an $\epsilon/2$-isometry and vice versa. A good place to read about this is [here](http://books.google.com/books?id=afnlx8sHmQIC&printsec=frontcover&dq=course+in+metric+geometry&source=bl&ots=JwC08A-Ze6&sig=YdYs4tF4UPJXqZJiYUexjWyKHt8&hl=en&ei=WjCSTZ34MMKnhAeC4NiEDw&sa=X&oi=book_result&ct=result&resnum=6&ved=0CD4Q6AEwBQ#v=onepage&q&f=false)




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Thus, this should give you some sort of condition on how close two compact manifolds can be in the Gromov-Hausdorff topology, if you know their spectrum. Rescale them so that $Ric \geq -(n-1)$, and then applying Theorem 7.11 you can get a lower bound on $\epsilon$. I'm not sure how explicit the $\delta$ is in their proof, however, if this is important to you.


