Nodal intersections and geometric control

We prove that the number of nodal points on an S-good real analytic curve C of a sequence S of Laplace eigenfunctions ∂_jof eigenvalue -λ²_jof a real analytic Riemannian manifold (M, g) is bounded above by A_g,Cλ_j. Moreover, we prove that the codimension- two Hausdorff measure H^m-2(N_∂λ∩H) of nodal intersections with a connected, irreducible real analytic hypersurface H ⊂ M is ≤ Ag;H λ_j. The S-goodness condition is that the sequence of normalized logarithms 1/λj log |∂j|²does not tend to -∞ uniformly on C, resp. H. We further show that a hypersurface satisfying a geometric control condition is S-good for a density one subsequence of eigenfunctions. This gives a partial answer to a question of Bourgain-Rudnick about hypersurfaces on which a sequence of eigenfunctions can vanish. The partial answer characterizes hypersurfaces on which a positive density sequence can vanish or just have L²norms tending to zero.