# Relationship between spectrum geometry and almost-isometry

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.

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I don't think you're asking the right question: the theory of isospectral manifolds generally focuses on closed manifolds, but any two compact metric spaces are automatically quasi-isometric. Quasi-isometries don't care at all about local structure - they are best adapted to large scale problems involving non-smooth spaces. So it's hard to imagine what the sort of connection that you're asking about would look like. –  Paul Siegel Mar 29 '11 at 11:19
Sorry, I failed to express my idea. My focus is about local structure, I misuse the concept of quasi-isometry. However I did not found a way to describe which two manifold are "almost" isometric. For example, the bounding surfaces of a non-rigid object may be "near" isometric of different poses. Their spectrum data seems be continuously change with respect to pose change. The deformation of a metric space will have corresponding change of their spectrum data. For two metric, considering their map $f$. inf{e: |d(x,y)-d(f(x),f(y)|< e} estimate compute min w.r.t $f$ –  bobye Mar 29 '11 at 11:42

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.

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, which also links to some interesting looking work by Lott about how the same question with the laplacian on $p$-forms.

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

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.

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The sphere, the torus and the compact surfaces of genus $g>1$ have very different Laplace-Beltrami operators. However, since all of them are compact are quasi-isometric to a point. As Paul mentioned, the local properties of the manifold are irrelevant to the quasi-isometries. Quasi-isometries only capture the large scale phenomenon of the manifold.