# The Stability of the First Neumann Laplacian Eigenfunction Under Domain Deformations and Applications

## TL;DR

This paper investigates the stability of the first Neumann Laplacian eigenfunction under domain deformations, providing theoretical insights and applications in geophysical data analysis.

## Contribution

It offers a rigorous analysis of eigenfunction stability under domain deformations and demonstrates its relevance to manifold learning and geophysical applications.

## Key findings

- First eigenfunction stability depends mainly on domain length in tall thin domains.
- Theoretical proof of eigenfunction stability under domain diffeomorphisms.
- Application to geophysical interpretation enhances understanding of domain deformation effects.

## Abstract

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.