Semicontinuity of Eigenvalues under Flat Convergence in Euclidean Space

TL;DR

This paper proves the upper semicontinuity of Neumann and Dirichlet eigenvalues of submanifolds under flat convergence, with conditions on volume and boundaries, extending previous work on eigenvalue behavior in geometric convergence.

Contribution

It establishes semicontinuity of eigenvalues under flat convergence with necessary boundary and volume conditions, expanding understanding beyond metric measure convergence.

Findings

Neumann eigenvalues are upper semicontinuous under flat convergence.

Dirichlet eigenvalues are semicontinuous with boundary conditions.

Continuity of eigenvalues is impossible under weaker hypotheses.

Abstract

Recall that Federer-Fleming defined the notion of flat convergence of submanifolds of Euclidean space to solve the Plateau problem. Here we prove the upper semicontinuity of Neumann eigenvalues of the submanifolds when they converge in the flat sense without losing volume. With an additional condition on the boundaries of the submanifolds we prove the Dirichlet eigenvalues are semicontinuous as well. We show this additional boundary condition is necessary as well as the condition that the volumes converge to the volume of the limit submanifold. As an application of our theorems we see that the Dirichlet and Neumann eigenvalues of a sequence of surfaces with a common smooth boundary curve approaching the solution to the Plateau problem are upper semicontinuous. This work is built upon Fukaya's study of the metric measure convergence of Riemannian manifolds. One may recall that Cheeger-Colding proved continuity of the eigenvalues when manifolds with uniform lower Ricci curvature bounds converge in the metric measure sense. While they obtain continuity, here, we produce an example demonstrating that continuity is impossible to obtain with our weaker hypothesis.