Spectral convergence of Riemannian manifolds, II

Abstract

We prove a precompactness theorem concerning the spectral distance on the set of isometry classes of compact Riemannian manifolds and study the comple- tion of a precompact family. Introduction. For a compact connected Riemannian manifold M=(M,g), we denote by pM(t, x, y) the heat kernel of the Laplace operator of M with respect ot the normarized Riemannian measure μM (= dvg/Vol(M)). Given two compact connected Riemannian manifolds M and N, a mapping /: M->N is called an e-spectral ap- proximation if it satisfies e-{t + llt) \pM(U x, y)~PN(t, f(x)9 Ry)) I 0 and x,yeM. The spectral distance SD(M, N) between M and N is by definition the lower bound of the positive numbers e such that there exist e-spectral approximations /: M-^N and h: N^M. The distance SD gives a uniform structure on the set Mc of isometry classes of compact connected Riemannian manifolds. Riemannian manifolds are considered as metric spaces endowed with Riemannian distances. From this point of view, the set Mc has another uniform structure introduced by Gromov (18), called the Hausdorff distance HD. In (18), the conditions for a family of Jic to be HD-precompact are described and it is shown that the boundaries of such a family consist of certain metric spaces, called length spaces. This decade has seen intensive activities around the convergence theory of Riemannian manifolds with respect to the Gromov-Hausdorff distance. These includes some works from the viewpoint of spectral geometry, for instance, (14), (4), and (23). In (25), motivated by these results, we introduced the spectral distance SD mentioned above and discussed some basic properties of the distance on a set of compact connected Riemannian manifolds of the same dimension with diameters uniformly bounded from above and Ricci curvatures uniformly bounded from below. In the present paper, we are concerned with a certain precompact family of Jίc and its compactification with respect to the spectral distance. More precisely, the main results are stated as follows.