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1 results

  • 10 - Neumann Domains on Graphs and Manifolds

  • Edited by Matthias Keller, Universität Potsdam, Germany, Daniel Lenz, Universität Potsdam, Germany, Radoslaw K. Wojciechowski
  • Book:
    Analysis and Geometry on Graphs and Manifolds
    Published online:
    14 August 2020
    Print publication:
    20 August 2020, pp 203-249

  • Summary

    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.