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.
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Acknowledgements
I would like to thank Emanuel Milman for proposing an idea that has led to simplification of the proof of Theorem 1.3 and Corollary 1.4. I am grateful to Sasha Logunov for vivid explanations on harmonic analysis, and to David Jerison, Misha Sodin and Steve Zelditch for their interest and for their remarks on an earlier version of this text. Supported in part by the Israeli Science Foundation (ISF) (Grant Number 765/19).
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Klartag, B. Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula. Geom Dedicata 208, 13–29 (2020). https://doi.org/10.1007/s10711-019-00507-4
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DOI: https://doi.org/10.1007/s10711-019-00507-4