2014/09/16 by Ram Band, Band, Ram, David Fajman +1 · 1 citation
Computer Science · Mathematics · #35Pxx #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Spectral Theory (math.SP) #math.SP #msc:35Pxx
paper · pdf · doi:10.48550/arxiv.1409.4793
openalex publication_date 2014/09/16 · arxiv created 2015/09/09 · arxiv updated 2015/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction - those which are connected to saddle points. These give rise to Neumann domains. We establish complementary definitions for Neumann domains and Neumann lines and use basic Morse homology to prove their fundamental topological properties. We study the eigenfunction restrictions to these domains. Their zero set, critical points and spectral properties allow to discuss some aspects of counting the number of Neumann domains and estimating their geometry.