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2021 Restriction of toral eigenfunctions to totally geodesic submanifolds
Xiaoqi Huang, Cheng Zhang
Anal. PDE 14(3): 861-880 (2021). DOI: 10.2140/apde.2021.14.861

Abstract

We estimate the L2 norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus 𝕋d, d≥2. We reduce getting correct bounds to counting lattice points in the intersection of some ν-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain–Rudnick on L2-restriction estimates for rational hyperplanes. On 𝕋2, we prove the uniform L2 restriction bounds for closed geodesics. On 𝕋3, we obtain explicit L2 restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge.

Citation

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Xiaoqi Huang. Cheng Zhang. "Restriction of toral eigenfunctions to totally geodesic submanifolds." Anal. PDE 14 (3) 861 - 880, 2021. https://doi.org/10.2140/apde.2021.14.861

Information

Received: 27 February 2019; Revised: 24 September 2019; Accepted: 21 November 2019; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/apde.2021.14.861

Subjects:
Primary: 11P21, 35P20, 58J50

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.14 • No. 3 • 2021
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