2019/10/07 by Huang, Shaosai, Taskent, Selin
#53C21 #58J60 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1910.02580
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 L2-average sense. This generalizes an estimate of Fukaya in the case of collapsing with bounded diameter and sectional curvature.