TY - JOUR

T1 - Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula

AU - Klartag, Bo’az

N1 - Publisher Copyright:
© 2019, Springer Nature B.V.

PY - 2020/10/1

Y1 - 2020/10/1

N2 - 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.

AB - 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.

KW - Laplace eigenfunctions

KW - Nodal sets

KW - Weighted minimal surfaces

UR - http://www.scopus.com/inward/record.url?scp=85077167564&partnerID=8YFLogxK

U2 - 10.1007/s10711-019-00507-4

DO - 10.1007/s10711-019-00507-4

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85077167564

SN - 0046-5755

VL - 208

SP - 13

EP - 29

JO - Geometriae Dedicata

JF - Geometriae Dedicata

IS - 1

ER -