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.

我们估计 ${\mathrm{大号}}^{\mathrm{2个}}$ 标准平面环上Laplace–Beltrami算子的本征函数的全测地子流形的限制范数 ${\mathbb{𝕋}}^d$， $d\ge \mathrm{2个}$。我们减少了对某些点的交点中的晶格点进行计数的正确边界$\nu$-球体上的横向带。此外，我们证明了任意余维的有理全测地子流形的正确边界。特别是，我们验证了布尔加因-拉德尼克（Bourgain-Rudnick）的猜想${\mathrm{大号}}^{\mathrm{2个}}$-有理超平面的限制估计。上 ${\mathbb{𝕋}}^{\mathrm{2个}}$，我们证明制服 ${\mathrm{大号}}^{\mathrm{2个}}$封闭大地测量学的限制范围。上${\mathbb{𝕋}}^3$，我们得到显式 ${\mathrm{大号}}^{\mathrm{2个}}$ 完全测地子流形的限制估计，这改善了Burq–Gérard–Tzvetkov，Hu和Chen–Sogge的相应结果。