Topological Properties of Neumann Domains

TL;DR

This paper explores the topological and spectral properties of Neumann domains, partitions of a manifold induced by Laplacian eigenfunctions, revealing their fundamental characteristics and potential for counting and geometric estimation.

Contribution

It introduces a rigorous topological framework for Neumann domains, using Morse homology, and analyzes their spectral and geometric properties, advancing understanding beyond nodal domains.

Findings

Neumann domains are characterized by their topological properties using Morse homology.

Eigenfunction restrictions to Neumann domains reveal critical points and zero sets.

The paper discusses methods for counting Neumann domains and estimating their geometry.

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.