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 phi(j) of eigenvalue -lambda(2)(j) of a real analytic Riemannian manifold (M, g) is bounded above by A(g,C) lambda(j). Moreover, we prove that the codimension-two Hausdorff measure Hm-2 (N-phi lambda boolean AND H) of nodal intersections with a connected, irreducible real analytic hypersurface H subset of M is <= A(g,) (H) lambda(j). The S -goodness condition is that the sequence of normalized logarithms 1/lambda(j) log vertical bar phi(j)vertical bar(2) does not tend to -infinity 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 L-2 norms tending to zero.