Nodal Sets of Laplace Eigenfunctions: Estimates of the Hausdorff Measure in Dimensions Two and Three

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.

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“…Let us also mention that in dimension n = 2, one can improve 3/4 from Donnelly-Fefferman's bound by a tiny ε ( [57]):…”

Section: Yau's Conjecturementioning

confidence: 99%

“…Remark. The recent combinatorial argument [57] shows that there exists ε > 0 such that for any closed surface M the eigenfunctions ϕ λ on M satisfy…”

Section: Estimate Of the Length Of Nodal Lines In Terms Of The Doublimentioning

confidence: 99%

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Logunov

Malinnikova

2018

Current Developments in Mathematics

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“…for every j ≥ 1. Yau's conjecture was proved by Donnelly and Fefferman in [DF88] for real analytic manifolds, and the lower bound was established by Logunov and Malinnikova [Log16a,Log16b,LM16] for the general case.…”

Section: Introduction 1background and Motivationsmentioning

confidence: 95%

Random nodal lengths and Wiener chaos

Rossi¹

2019

Probabilistic Methods in Geometry, Topology and Spectral Theory

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In this survey we collect some of the recent results on the "nodal geometry" of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace eigenfunctions on the torus (arithmetic random waves) and on the sphere (random spherical harmonics). We give some insight on both Berry's cancellation phenomenon and the nature of nodal length second order fluctuations (non-Gaussian on the torus and Gaussian on the sphere) in terms of chaotic components. Finally we consider the general case of monochromatic random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian surface with frequencies from a short interval, whose scaling limit is Berry's Random Wave Model. For the latter we present some recent results on the asymptotic distribution of its nodal length in the high energy limit (equivalently, for growing domains).

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“…For non-analytic manifolds the best-known upper estimate in dimension n = 2 was H 1 ({ϕ λ = 0}) ≤ Cλ 3/4 due to Donnelly and Fefferman ([6]), different proof for the same bound was given by Dong ([4]). Recently this bound was refined to Cλ 3/4−ε in [12]. In higher dimensions the estimate H n−1 ({ϕ λ = 0}) ≤ Cλ C √ λ by Hardt and Simon ( [9]) was the only known upper bound till now.…”

Section: Preliminariesmentioning

confidence: 99%