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Averages of Eigenfunctions Over Hypersurfaces

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Abstract

Let (M, g) be a compact, smooth, Riemannian manifold and \({\{ \phi_h \}}\) an L2-normalized sequence of Laplace eigenfunctions with defect measure \({\mu}\). Let H be a smooth hypersurface with unit exterior normal \(\nu\). Our main result says that when \(\mu\) is not concentrated conormally to H, the eigenfunction restrictions to H satisfy

$$\int_H \phi_h d\sigma_H = o(1) \quad {\rm and} \quad \int_H h D_{\nu} \phi_h d\sigma_H = o(1),$$

\({h \to 0^+}\).

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA

    Yaiza Canzani

  2. Department of Mathematics, Stanford University, Stanford, CA, USA

    Jeffrey Galkowski

  3. Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada

    John A. Toth

Authors
  1. Yaiza Canzani
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  2. Jeffrey Galkowski
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  3. John A. Toth
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Correspondence to Jeffrey Galkowski.

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Communicated by J. Marklof

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Canzani, Y., Galkowski, J. & Toth, J.A. Averages of Eigenfunctions Over Hypersurfaces. Commun. Math. Phys. 360, 619–637 (2018). https://doi.org/10.1007/s00220-017-3081-9

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  • DOI: https://doi.org/10.1007/s00220-017-3081-9

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