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.
- $|d(x,y)-d(f(x),f(y))|<\varepsilon$ for $x,y\in X$
- 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.
