# Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula

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## Abstract

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 language English 13-29 17 Geometriae Dedicata 208 1 https://doi.org/10.1007/s10711-019-00507-4 Published - 1 Oct 2020 Yes

## Keywords

• Laplace eigenfunctions
• Nodal sets
• Weighted minimal surfaces

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