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.

流形学习方法的鲁棒性通常取决于假设的基础域变形下的Neumann Laplacian特征函数的稳定性。实际上，许多流形学习方法都基于在流形上近似Neumann拉普拉斯特征函数的假设，该流形是假设数据的基础，而数据是通过失真源查看的。在本文中，我们研究了第一个Neumann Laplacian特征函数关于微分同域变形的稳定性。特别是，我们对第一个本征函数在高个薄域上的稳定性感兴趣，直觉上，第一个Neumann Laplacian本征函数应仅取决于沿该域的长度。我们证明了该声明的严格版本，并将其应用于地球物理解释中的机器学习问题。