Topological Properties of Neumann Domains

Abstract

Abstract. 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.

Neumann Domains on Quantum Graphs

“…Remark 2.5. As the goal of this paper is to define and analyze Neumann domains on graphs, we should mention another class of eigenfunctions used when considering Neumann domains on manifolds [5,11,10,30,12], which is Morse eigenfunctions. A Morse eigenfunction is such that at no point both the function and its derivative vanish.…”

“…To the best of our knowledge, this is the first work on Neumann domains on graphs 2 . Even on manifolds, Neumann domains are a very recent topic of research within spectral theory and is currently mentioned only in [37,30,12,10,5]. Partial results of the current paper were already announced in [5] which reviews the Neumann domain research on manifolds and on graphs.…”

The Spectral Position of Neumann Domains on the Torus

“…The above defines a partition of the manifold M , which we call the Neumann partition. Indeed, it is not hard to show that M equals the disjoint union of all Neumann domains and the Neumann line set (under the assumption that N (f ) = ∅, see [BF16,Proposition 1.3]). Figure 1.1 depicts the Neumann partition of a particular eigenfunction on the flat torus.…”

“…The spectral position is a key notion for Neumann domains. Finding its value is a great challenge and is of major importance in studying Neumann domains and their properties [BF16,BET20,ABBE20]. As a comparison, the similar notion for a nodal domain is trivial: if Ξ is a nodal domain of f , then f | Ξ is the first eigenfunction of the Dirichlet Laplacian on Ξ.…”

On the hot spots of quantum graphs

“…These same motivations remain valid in the case of quantum graphs, where comparatively little seems to be known, at least in terms of the profile of the eigenfunctions: some work has been done constructing so-called landscape functions to control their size [34,35], and relatively recently the concept of Neumann domains of the eigenfunctions, the regions separated by critical points of the eigenfunctions, was introduced and is now being studied [5,6,8,13]. But to date the "hot and cold spots" of a quantum graph do not seem to have received direct attention, a preliminary note of the current authors excluded [44].…”