  The robustness of manifold learning methods is often predicated on the
stability of the Neumann Laplacian eigenfunctions under deformations of the
assumed underlying domain. Indeed, many manifold learning methods are based on
approximating the Neumann Laplacian eigenfunctions on a manifold that is
assumed to underlie data, which is viewed through a source of distortion. In
this paper, we study the stability of the first Neumann Laplacian eigenfunction
with respect to deformations of a domain by a diffeomorphism. In particular, we
are interested in the stability of the first eigenfunction on tall thin domains
where, intuitively, the first Neumann Laplacian eigenfunction should only
depend on the length along the domain. We prove a rigorous version of this
statement and apply it to a machine learning problem in geophysical
interpretation.
