Neumann Domains on Graphs and Manifolds

Lior Alon, Ram Band, Michael Bersudsky, Sebastian Egger

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


A Laplacian eigenfunction on a manifold or a metric graph imposes a natural partition of the manifold or the graph. This partition is determined by the gradient vector field of the eigenfunction (on a manifold) or by the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. Their counterparts are the well-known nodal domains. This paper reviews the subject of Neumann domains, as appears in recent publications 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.

Original languageEnglish
Title of host publicationAnalysis and Geometry on Graphs and Manifolds
Number of pages47
ISBN (Electronic)9781108615259
StatePublished - 1 Jan 2020


  • Laplacian eigenfunctions
  • Morse-Smale complexes
  • Neumann domains
  • Neumann lines
  • Nodal domains
  • Quantum graph

All Science Journal Classification (ASJC) codes

  • General Mathematics


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