Restriction of toral eigenfunctions to totally geodesic submanifolds

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.