Nodal sets of Laplace eigenfunctions: Polynomial upper estimates of the Hausdorff measure

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Abstract

Let M be a compact C∞-smooth Riemannian manifold of dimension n, n ≥ 3, and let ψλ: ΔMψλ + λψλ = 0 denote the Laplace eigenfunction on M corresponding to the eigenvalue λ. We show that Hn-1((ψλ = 0)) ≤ Cλα where α > 1/2 is a constant, which depends on n only, and C > 0 depends on M . This result is a consequence of our study of zero sets of harmonic functions on C∞-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.

Original languageEnglish (US)
Pages (from-to)221-239
Number of pages19
JournalAnnals of Mathematics
Volume187
Issue number1
DOIs
StatePublished - Jan 1 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Doubling index
  • Frequency
  • Harmonic functions
  • Laplace eigenfunctions
  • Nodal sets
  • Yau's conjecture

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