## Spectral convergence under bounded Ricci curvature. (arXiv:1510.05349v2 [math.DG] UPDATED)

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian
manifolds with a uniform bound of Ricci curvature, we establish two spectral
convergence. One of them is on the Hodge Laplacian acting on differential
one-forms. The other is on the connection Laplacian acting on tensor fields of
every type, which include all differential forms. These are generalizations of
Cheeger-Colding's spectral convergence of the Laplacian acting on functions to
the cases of tensor fields and differential forms. As a corollary we show the
upper semicontinuity of the first Betti numbers with respect to the
Gromov-Hausdorff topology, and give the equivalence between the continuity of
them and the existence of a uniform spectral gap. We also define the Riemannian
curvature tensor, the Ricci curvature, and the scalar curvature of the limit
space, and show their properties. In particular the Ricci curvature coincides
with the difference between the Hodge Laplacian and the connection Laplacian,
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