Nonexistence of invariant nodal line and improved $L^2$ restriction bounds for Neumann data on negatively curved surface

Abstract

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 \cite{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.

The results of this paper are false. There is a counter example: the odd eigenfunctions vanish on a curve which is fixed by an isometric involution