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- W2947995679 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." @default.
- W2947995679 created "2019-06-07" @default.
- W2947995679 creator A5059794784 @default.
- W2947995679 date "2019-05-28" @default.
- W2947995679 modified "2023-10-17" @default.
- W2947995679 title "Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula" @default.
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