Nodal intersections and geometric control

John A. Toth, Steve Zelditch

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We prove that the number of nodal points on an S-good real analytic curve C of a sequence S of Laplace eigenfunctions ∂jof eigenvalue -λ2jof a real analytic Riemannian manifold (M, g) is bounded above by Ag,Cλj. Moreover, we prove that the codimension- two Hausdorff measure Hm-2(N∂λ∩H) of nodal intersections with a connected, irreducible real analytic hypersurface H ⊂ M is ≤ Ag;H λj. The S-goodness condition is that the sequence of normalized logarithms 1/λj log |∂j|2does not tend to -∞ uniformly on C, resp. H. We further show that a hypersurface satisfying a geometric control condition is S-good for a density one subsequence of eigenfunctions. This gives a partial answer to a question of Bourgain-Rudnick about hypersurfaces on which a sequence of eigenfunctions can vanish. The partial answer characterizes hypersurfaces on which a positive density sequence can vanish or just have L2norms tending to zero.

Original languageEnglish (US)
Pages (from-to)345-393
Number of pages49
JournalJournal of Differential Geometry
Volume117
Issue number2
DOIs
StatePublished - Feb 2021

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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