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By Cheeger-Colding's almost splitting theorem, if a domain in a Ricci flatmanifold is pointed-Gromov-Hausdorff close to a lower dimensional Euclideandomain, then there is a harmonic almost splitting map. We show that anyeigenfunction of the Laplace operator is almost constant along the fibers ofthe almost splitting map, in the $L^2$-average sense. This generalizes anestimate of Fukaya in the case of collapsing with bounded diameter andsectional curvature.

Small fiberwise oscillation of the eigenfunctions of collapsing Einstein manifolds