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 $\gamma$ be a closed smooth curve with unit exterior normal $\nu$. We can show that $\| h \partial_\nu u_{h} \|_{L^2(\Gamma)}=o(1)$ if $\{u_h\}$ is tangentially concentrated with respect to $\gamma$. As a key ingredient of the proof, we give a detailed analysis of the $L^2$ norms over $\gamma$ of the Neumann data $h\partial_\nu u_h$ when mircolocalized away the cotangential direction.