On the nodal domain count under metric perturbations, Courant sharpness and boundary intersections

2025/07/07 by Mukherjee, Mayukh, Maji, Saikat, Saha, Soumyajit
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2507.04928

Abstract

In this note, we investigate the effects of metric perturbations of closed surfaces on the number of nodal domains. In particular, we show that the nodal domain count for Laplace eigenfunctions is (locally) maximised for non-generic metrics. We also investigate localised perturbations and showcase some interesting applications regarding Courant sharp metrics on closed surfaces and associated boundary nodal-data prescription problems.

Citations

• Nodal Sets of Laplacian Eigenfunctions with an Eigenvalue of Multiplicity 2
• The inner radius of nodal domains in high dimensions
• Nodal Set Openings on Perturbed Rectangular Domains
• On the effects of small perturbation on low energy Laplace eigenfunctions
• Nodal sets of Laplace eigenfunctions under small perturbations
• Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure
• Eigenfunctions with prescribed nodal sets
• Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces
• Bounds on the multiplicity of eigenvalues for fixed membranes

Related

• Some remarks on nodal geometry in the smooth setting
• Pleijel nodal domain theorem in non-smooth setting
• Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces
• Coarse nodal count and topological persistence
• The size of the nodal sets for the eigenfunctions of the smooth laplacian