Neumann Domains on Graphs and Manifolds

TL;DR

This paper reviews the concept of Neumann domains, a partition based on eigenfunctions' gradient fields or extremal points, comparing results on manifolds and graphs, and discusses open questions in the field.

Contribution

It provides a comprehensive review of Neumann domains on manifolds and graphs, highlighting recent results and open problems in the study of eigenfunction partitions.

Findings

Neumann domains form natural partitions related to eigenfunctions.

Comparison between results on manifolds and graphs reveals similarities and differences.

Open questions and conjectures highlight future research directions.

Abstract

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. This paper reviews the subject, as appears in a few recent works and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.