Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula
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Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula Bo’az Klartag1 Received: 14 June 2019 / Accepted: 16 December 2019 © Springer Nature B.V. 2019
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. Keywords Laplace eigenfunctions · Weighted minimal surfaces · Nodal sets Mathematics Subject Classification 65N25
1 Introduction Consider an eigenfunction f of the Laplacian on a compact, n-dimensional, C ∞ -smooth Riemannian manifold M. Let μ be a non-zero finite, Borel measure on M, and let X be a random point in M, distributed according to the probability measure that is proportional to μ. The density function of the real-valued random variable f (X ) is referred to as the value distribution density of f under μ. One of the first measures to look at is of course the uniform measure σ , the Riemannian volume measure. The random wave conjecture of Berry [4] suggests that under ergodicity assumptions, in the generic case “a random wave is a random function”. That is, a Laplace eigenfunction corresponding to a large eigenvalue should have a value distribution density under σ that is approximately Gaussian. We refer to Jakobson, Nadirashvili and Toth [15] and to Zelditch [23] for background on eigenfunctions of the Laplacian on Riemannian manifolds.
B 1
Bo’az Klartag [email protected] Department of Mathematics, Weizmann Institute of Science, 76100 Rehovot, Israel
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Geometriae Dedicata
Let us discuss the example of the standard flat torus M = Rn /Zn , even though its geodesic flow is not ergodic. In this case, the eigenfunctions of the simplest form are perhaps f k (x1 , . . . , xn ) = sin(2πkx1 )
(k = 0, 1, 2 . . .),
(1)
defined for x = (x1 , . . . , xn ) ∈ Rn /Zn . Then f k = −4π 2 k 2 f k , and the eigenvalue −4π 2 k 2 tends to −∞ with k. Nevertheless, the value distribution density of f k under the Riemannian volume measure is independent of k, and it equals t →
1 √ π 1 − t2
for t ∈ (−1, 1).
(2)
This density function is unbounded, and its graph bears little resemblance to the graph of the √ 2 Gaussian density e−t /2 / 2π. In fact, the density function in (2) is not even unimodal with a maximum at the origin, but quite the opposite.
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