Small fiberwise oscillation of the eigenfunctions of collapsing Einstein manifolds

Abstract

By Cheeger-Colding's almost splitting theorem, if a domain in a Ricci flat manifold is pointed-Gromov-Hausdorff close to a lower dimensional Euclidean domain, then there is a harmonic almost splitting map. We show that any eigenfunction of the Laplace operator is almost constant along the fibers of the almost splitting map, in the L 2 -average sense. This generalizes an estimate of Fukaya in the case of collapsing with bounded diameter and sectional curvature.Here we recall that when the Riemannian manifold (M i , g) is sufficiently close to, in the Gromov-Hausdorff sense, another lower diemnsional Riemannian manifold (N, h), then the regularity assumptions (1.1) guarantee that there is a fibration Φ i : M i → N, which is also an almost Riemannian submersion. Moreover, the Φ i fibers, as embedded submanifolds in M i , are all homeomorphic