  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.
