Nodal sets of Laplace eigenfunctions: Proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture

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Abstract

Let u be a harmonic function in the unit ball B(0; 1) ⊂ ℝn, n ≥ 3, such that u(0) = 0. Nadirashvili conjectured that there exists a positive constant c, depending on the dimension n only, such that Hn-1((u = 0) (n-ary intersection) B) ≥ c. We prove Nadirashvili's conjecture as well as its counterpart on C∞-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjec- ture. Namely, we show that for any compact C∞-smooth Riemannian manifold M (without boundary) of dimension n, there exists c > 0 such that for any Laplace eigenfunction ψλ on M, which corresponds to the eigenvalue λ, the following inequality holds: c λ ≤ Hn-1((ψλ=0))).

Original languageEnglish (US)
Pages (from-to)241-262
Number of pages22
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

  • Dou- bling index
  • Frequency
  • Harmonic functions
  • Laplace eigenfunctions
  • Nodal sets
  • Yau's conjecture

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