Vol. 14, No. 3, 2021

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
This article is available for purchase or by subscription. See below.
Restriction of toral eigenfunctions to totally geodesic submanifolds

Xiaoqi Huang and Cheng Zhang

Vol. 14 (2021), No. 3, 861–880
Abstract

We estimate the ${L}^{2}$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus ${\mathbb{𝕋}}^{d}$, $d\ge 2$. We reduce getting correct bounds to counting lattice points in the intersection of some $\nu$-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 ${L}^{2}$-restriction estimates for rational hyperplanes. On ${\mathbb{𝕋}}^{2}$, we prove the uniform ${L}^{2}$ restriction bounds for closed geodesics. On ${\mathbb{𝕋}}^{3}$, we obtain explicit ${L}^{2}$ restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge.

PDF Access Denied

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/apde

We have not been able to recognize your IP address 3.238.204.31 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords
eigenfunction estimates, restriction, geodesic
Mathematical Subject Classification 2010
Primary: 11P21, 35P20, 58J50
Milestones
Received: 27 February 2019
Revised: 24 September 2019
Accepted: 21 November 2019
Published: 18 May 2021
Authors
 Xiaoqi Huang Department of Mathematics Johns Hopkins University Baltimore, MD United States Cheng Zhang Department of Mathematics Johns Hopkins University Baltimore, MD United States