Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula

Bo’az Klartag*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let M be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by σ. Let f: M→ R be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of f under σ is approximately Gaussian. Write μ for the measure whose density with respect to σ is | ∇ f| 2. We observe that the value distribution of f under μ admits a unimodal density attaining its maximum at the origin. Thus, in a sense, the zero set of an eigenfunction is the largest of all level sets. When M is a manifold with boundary, the same holds for Laplace eigenfunctions satisfying either the Dirichlet or the Neumann boundary conditions. Additionally, we prove a monotonicity formula for level sets of solid spherical harmonics, essentially by viewing nodal sets of harmonic functions as weighted minimal hypersurfaces.

Original languageEnglish
Pages (from-to)13-29
Number of pages17
JournalGeometriae Dedicata
Issue number1
StatePublished - 1 Oct 2020
Externally publishedYes


  • Laplace eigenfunctions
  • Nodal sets
  • Weighted minimal surfaces


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