时间：2024年7月9号，15:00-16:00

地点：2206

报告人：吴先超（武汉理工大学)

题目：Restriction bounds for Neumann Data on surface

摘要：Let $\{u_\lambda\}$ be a sequence of $L^2$-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. Firstly, we aim to provide a simple proof of an $L^p$ restriction bounds of the Neumann data $ \lambda^{-1} \partial_\nu u_{\lambda}\,\vline_\gamma$ along a unit geodesic $\gamma$. Using the $T$-$T^*$ argument one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is $O(\lambda^{-\frac{1}p+\frac{3}2})$. Moreover, this upper bound is shown to be optimal.

Secondly, we seek to prove the same restricted upper bound without the geodesic assumption. Furthermore we can obtain an improved result on an asymmetric curve if the surface is negatively curved. The novelty of the result induces the uniformly lower bounds for eigenfunctions on such surfaces and shows the nonexistence of invariant asymmetric nodal lines.