Nonexistence of invariant nodal line and improved L² restriction bounds for Neumann data on negatively curved surface

The problem of obtaining the lower bounds on the restriction of Laplacian eigenfunctions to hypersurfaces inside a compact Riemannian manifold $(M,g)$ is challenging and has been attempted by many authors {BR, GRS, Jun, ET}. This paper aims to show that if $(M,g)$ is assumed to be a negatively curved surface then one can get the corresponding restricted lower bounds, as well as quantitative improvement of restricted bounds for Neumann data.