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February 2021 Nodal intersections and geometric control
John A. Toth, Steve Zelditch
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J. Differential Geom. 117(2): 345-393 (February 2021). DOI: 10.4310/jdg/1612975018

Abstract

We prove that the number of nodal points on an $\mathcal{S}$-good real analytic curve $\mathcal{C}$ of a sequence $\mathcal{S}$ of Laplace eigenfunctions $\varphi_j$ of eigenvalue $-\lambda^2_j$ of a real analytic Riemannian manifold $(M, g)$ is bounded above by $A_{g , \mathcal{C}} \lambda_j$. Moreover, we prove that the codimension-two Hausdorff measure $\mathcal{H}^{m-2} (\mathcal{N}_{\varphi \lambda} \cap H)$ of nodal intersections with a connected, irreducible real analytic hypersurface $H \subset M$ is $\leq A_{g, H} \lambda_j$. The $\mathcal{S}$-goodness condition is that the sequence of normalized logarithms $\frac{1}{\lambda_j} \operatorname{log} {\lvert \varphi_j \rvert}^2$ does not tend to $-\infty$ uniformly on $\mathcal{C}$, resp. $H$. We further show that a hypersurface satisfying a geometric control condition is $\mathcal{S}$-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 $L^2$ norms tending to zero.

Funding Statement

The research of J.T. was partially supported by NSERC Discovery Grant # OGP0170280, an FRQNT Team Grant and the French National Research Agency project Gerasic-ANR-13-BS01-0007-0.
The research of S.Z. was partially supported by NSF grant # DMS-1541126.

Citation

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John A. Toth. Steve Zelditch. "Nodal intersections and geometric control." J. Differential Geom. 117 (2) 345 - 393, February 2021. https://doi.org/10.4310/jdg/1612975018

Information

Received: 30 August 2017; Published: February 2021
First available in Project Euclid: 10 February 2021

Digital Object Identifier: 10.4310/jdg/1612975018

Rights: Copyright © 2021 Lehigh University

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Vol.117 • No. 2 • February 2021
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