On the eigenforms of compact stratified spaces

Abstract

Let X be a compact Thom–Mather stratified pseudomanifold, and let M be the regular part of X endowed with an iterated metric. In this paper, we prove that if the curvature operator of M is bounded, then the \(L^2\) harmonic space of M is finite dimensional. Next we consider the absolute eigenvalue problems of the Hodge Laplacian of a sequence of compact domains \(\Omega _j\) converging to M. We prove that when the curvature operator of M is bounded, the eigenvalues of \(\Omega _j\) converge to eigenvalues of M, and the eigenforms of \(\Omega _j\) converge to eigenforms of M in the Sobolev norm. This generalizes Chavel and Feldman’s theorem in Chavel and Feldman (J Funct Anal 30:198-222, 1978) from compact manifolds to compact pseudomanifolds and from functions to differential forms. Then, we apply our results to \(L^2\)-chomology. We will give a correspondence between boundary cohomology and \(L^2\)-cohomology.