Relationship between spectrum geometry and almost-isometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:13:42Z http://mathoverflow.net/feeds/question/59944 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59944/relationship-between-spectrum-geometry-and-almost-isometry Relationship between spectrum geometry and almost-isometry bobye 2011-03-29T05:02:31Z 2011-03-29T19:28:21Z <p>Sorry, I misuse the concept of quasi-isometry, I mean almost isometry(also called a Hausdorff approximation).</p> <p>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?</p> <p>Almost isometry say for two metric space(Riemann manifold). there exist $\varepsilon$ and $f: X\rightarrow Y$ s.t.</p> <ol> <li>$|d(x,y)-d(f(x),f(y))|&lt;\varepsilon$ for $x,y\in X$</li> <li>for any point $y\in Y$, there exists an $x\in X$ s.t. $d(f(x),y)&lt;\varepsilon$</li> </ol> <p>the question is: Given two riemann manifold, how to check almost isometry and estimate inf ${\varepsilon}$ from spectrum data.</p> http://mathoverflow.net/questions/59944/relationship-between-spectrum-geometry-and-almost-isometry/59966#59966 Answer by ght for Relationship between spectrum geometry and almost-isometry ght 2011-03-29T11:38:53Z 2011-03-29T11:38:53Z <p>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.</p> <p>On the other hand, if I recall correctly if your manifold has the linear isoperimetric property (which is captured in the spectrum of the Laplacian) so does a quasi-isometric manifold. </p> http://mathoverflow.net/questions/59944/relationship-between-spectrum-geometry-and-almost-isometry/60013#60013 Answer by Otis Chodosh for Relationship between spectrum geometry and almost-isometry Otis Chodosh 2011-03-29T19:22:50Z 2011-03-29T19:28:21Z <p>The above comments are still mostly valid with $\epsilon$-isometries:</p> <p>if $(M,g)$ and $(N,g)$ are Riemannian manifolds with diameters less than $D$, then</p> <p>$f:M\to N$, $x\mapsto n_0$ for some $n_0\in N$ has</p> <p>$$|d_N(f(x),f(x')) - d_M(x,x')| = d_M(x,x') \leq D$$</p> <p>and for all $y \in N$, because $$d_N(y,n_0) \leq D$$</p> <p>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 <em>is</em> an interesting idea, but some sort noncompactness plays a big role in things, as should be obvious from all of these answers.</p> <hr> <p>On the other hand, $\epsilon$-isometries and the spectrum are certainly related in some ways, particularly with lower Ricci bounds:</p> <p>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. </p> <p>For example, their Theorem 7.11 says:</p> <blockquote> <blockquote> <p>For $M_1^n$, $M_2^n$, Riemannian manifolds satisfying $$Ric \geq -(n-1)$$ and $$diam(N_1^n) \leq d &lt; \infty$$ Then for all $N &lt; \infty$ and $\epsilon > 0$ there is a $\delta(n,d,\epsilon,N) > 0$ such that if $$d_{GH}(M_1^n,M_2^n) &lt; \delta$$ then for $j\leq N$, we have that $|\lambda_{j,1} - \lambda_{j,2}| &lt;\epsilon$, where $\lambda_{i,k}$ is the $i$-th eigenvalue on the $M_k$.</p> </blockquote> </blockquote> <p>You can get more information about this <a href="http://arxiv.org/abs/math/0612107" rel="nofollow">here</a>, which also links to some interesting looking work by Lott about how the same question with the laplacian on $p$-forms.</p> <hr> <p>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 $&lt;\epsilon$ if there exists an $\epsilon/2$-isometry and vice versa. A good place to read about this is <a href="http://books.google.com/books?id=afnlx8sHmQIC&amp;printsec=frontcover&amp;dq=course+in+metric+geometry&amp;source=bl&amp;ots=JwC08A-Ze6&amp;sig=YdYs4tF4UPJXqZJiYUexjWyKHt8&amp;hl=en&amp;ei=WjCSTZ34MMKnhAeC4NiEDw&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=6&amp;ved=0CD4Q6AEwBQ#v=onepage&amp;q&amp;f=false" rel="nofollow">here</a></p> <hr> <p>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.</p>