Improvements in $L^2$ Restriction bounds for Neumann Data along closed curves

Abstract

We seek to improve the restriction bounds of Neumann data of Laplace eigenfunctions $u_h$ by studying the $L^2$ restriction bounds of Neumann data and their $L^2$ concentration as measured by defect measures. Let $γ$ be a closed smooth curve with unit exterior normal $ν$. We can show that $\| h \partial_νu_{h} \|_{L^2(Γ)}=o(1)$ if $\{u_h\}$ is tangentially concentrated with respect to $γ$. As a key ingredient of the proof, we give a detailed analysis of the $L^2$ norms over $γ$ of the Neumann data $h\partial_νu_h$ when mircolocalized away the cotangential direction.