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 $\Vert h{\partial }_{\nu }{u}_h{\Vert }_{L^2(\Gamma )}=o(1)$ 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.

arXiv commentThe range of $\delta$ in Theorem 1 should be $\le 1/3$. And it still has some mistakes in the proof