#### Vol. 14, No. 3, 2021

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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.

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