Small fiberwise oscillation of the eigenfunctions of collapsing Einstein manifolds

TL;DR

This paper demonstrates that eigenfunctions on collapsing Ricci flat manifolds are nearly constant along fibers of an almost splitting map, extending previous estimates to more general collapsing scenarios.

Contribution

It generalizes Fukaya's estimate by showing eigenfunctions are almost constant along fibers in collapsing Ricci flat manifolds with lower dimensional limits.

Findings

Eigenfunctions are almost constant along fibers in collapsing manifolds.

Extension of Fukaya's estimate to more general collapsing cases.

Provides quantitative control of eigenfunctions in collapsing geometries.

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.