# Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula

## TL;DR

This paper investigates the distribution of Laplace eigenfunctions on Riemannian manifolds, showing a unimodal distribution centered at zero and establishing a monotonicity formula for level sets, extending to boundary conditions.

## Contribution

It demonstrates the unimodal value distribution of eigenfunctions under a specific measure and proves a new monotonicity formula for level sets of harmonic functions.

## Key findings

- Value distribution of eigenfunctions is unimodal and peaks at zero.
- Zero sets of eigenfunctions are the largest level sets.
- Monotonicity formula for level sets of spherical harmonics is established.

## Abstract

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $\sigma$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of $f$ under $\sigma$ is approximately Gaussian. Write $\mu$ for the measure whose density with respect to $\sigma$ is $|\nabla f|^2$. We observe that the value distribution of $f$ under $\mu$ 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.