Analysis & PDE ( IF 1.8 ) Pub Date : 2021-05-18 , DOI: 10.2140/apde.2021.14.861 Xiaoqi Huang , Cheng Zhang
We estimate the norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus , . 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 -restriction estimates for rational hyperplanes. On , we prove the uniform restriction bounds for closed geodesics. On , we obtain explicit restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge.
中文翻译:
局限本征函数对完全测地子流形的限制
我们估计 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 , 。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-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-有理超平面的限制估计。上 ,我们证明制服 封闭大地测量学的限制范围。上,我们得到显式 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。