# Restriction of toral eigenfunctions to totally geodesic submanifolds

## TL;DR

This paper establishes sharp $L^2$ restriction bounds for eigenfunctions of the Laplacian on flat tori, confirming conjectures and improving previous results for totally geodesic submanifolds of various codimensions.

## Contribution

It proves optimal $L^2$ restriction estimates for eigenfunctions on flat tori, including rational hyperplanes and geodesics, advancing understanding of eigenfunction behavior on these manifolds.

## Key findings

- Verified Bourgain-Rudnick conjecture for rational hyperplanes
- Established uniform bounds for geodesics on $ ext{T}^2$
- Improved restriction estimates on $ ext{T}^3$ for 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 $\mathbb{T}^d$, $d\ge2$. 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-G\'erard-Tzvetkov, Hu, Chen-Sogge.