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Embedding Riemannian manifolds by their heat kernel. (English) Zbl 0806.53044

Let \({\mathcal M}_{n,k,D}\) be the set of closed Riemannian manifolds of dimension \(n\) with Ricci curvature bounded below by \((n - 1)kg\), and diameter at most \(D\). Gromov showed this spaces has a precompact metric space structure. The authors use the heat kernel of the manifold to embed any Riemannian manifold of \({\mathcal M}_{n,k,D}\) into \(\ell^ 2\), the space of real valued square summable series. The curvature and diameter assumptions then yield that the image of \({\mathcal M}_{n,k,D}\) is bounded in a subspace \(h^ 1\) which embeds compactly into \(\ell^ 2\). Pulling back the Hausdorff distance between subsets of \(\ell^ 2\) then defines a distance on \({\mathcal M}_{n,k,D}\) which makes this space precompact and thereby gives another proof of Gromov’s result. The very elegant results of this paper rely upon very careful estimates of the heat kernel and on a result concerning \(C^ 0\) approximation of eigenfunctions which is of interest in its own right. The authors define a spectral distance and show the spectrum is continuous with respect to this distance.
Reviewer: P.Gilkey (Eugene)

MSC:

53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

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