Spectral convergence of Riemannian manifolds. II
Let \({\mathcal M}_c\) denote the isometry classes of compact connected Riemannian manifolds \(M\). Let \(\rho_M (t,x,y)\) be the heat kernel of the Laplace operator of \(M\) relative to the normalized measure \(\text{dvol}_M/\text{vol}(M)\). The spectral distance \(\text{SD}(M,N)\) measures the difference between \(\rho_M(t,x,y)\) and \(\rho_N(t,f(x),f(y))\) where the infimum is taken over all maps \(f: M \to N\) and is suitably normalized in \(t\); it gives a uniform structure to \({\mathcal M}_c\). The Hausdorff distance \(\text{HD} (M,N)\) is another uniform structure which was introduced by Gromov. The authors prove precompactness theorems concerning both SD and HD and they study the completion of a SD precompact family. [For part I see the authors, ibid. 46, No. 2, 147-179 (1994; Zbl 0814.53035)].
scientific article; zbMATH DE number 889576
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