Cheng Zhang

Johns Hopkins University

Talk Title: Restriction of toral eigenfunctions to totally geodesic submanifolds

Abstract: We estimate the L2 norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace-Beltrami operator on the standard flat torus Td, d?2. We reduce getting correct bounds to counting lattice points in the intersection of some ?-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 L2-restriction estimates for rational hyperplanes. On T2, we prove the uniform L2 restriction bounds for closed geodesics. On T3, we obtain explicit L2 restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq-Gérard-Tzvetkov, Hu, Chen-Sogge.