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Restriction of toral eigenfunctions to totally geodesic submanifolds
Analysis & PDE ( IF 1.8 ) Pub Date : 2021-05-18 , DOI: 10.2140/apde.2021.14.861
Xiaoqi Huang , Cheng Zhang

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 ν-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, 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 大号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 𝕋dd 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ν-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)的猜想大号2个-有理超平面的限制估计。上 𝕋2个,我们证明制服 大号2个封闭大地测量学的限制范围。上𝕋3 ,我们得到显式 大号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。

更新日期:2021-05-18
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