The 𝐿𝑝 restriction bounds for Neumann data on surface

Let { u λ } \{u_{\lambda}\} be a sequence of L 2 L^{2} -normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold ( M , g ) (M,g) . We seek to get an L p L^{p} restriction bound of the Neumann data λ − 1 ⁢ ∂ ν u λ | γ \lambda^{-1}\partial_{\nu}u_{\lambda}|_{\gamma} along a unit geodesic 𝛾. Using the 𝑇- 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 ⁢ ( λ − 1 p + 3 2 ) O(\lambda^{-\frac{1}{p}+\frac{3}{2}}) . The stationary phase theorem plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.