Nodal intersections and Geometric Control

Abstract

This article contains a generalization of the authors' results on numbers of nodal points of eigenfunctions on "good curves" in analytic plane domains (arXiv:0710.0101). The term `good' means that the $L^2$ norms of restrictions of eigenfunctions of eigenvalue $\lambda^2$ to the curve are bounded below by $e^{- C \lambda}$. In this article, the result is generalized to all real analytic Riemannian manifolds $(M, g)$ of any dimension $m$ without boundary. Moreover, a similar lower bound is given for the Hausdorff $m-2$ measure of the intersection of the nodal set with a good real analytic hypersurface. Most of the article is devoted to giving a dynamical or geometric control condition for `goodness' of a hypersurface. The conditions are that the hypersurface $H$ be asymmetric with respect to geodesics and that the flowout of the unit vectors with footpoint on $H$ have full measure in $S^*M. $ This gives a partial answer to a question of Bourgain-Rudnick of characterizing hypersurfaces $H$ on which a sequence of eigenfunctions vanishes. We show that under our conditions, a positive density sequence cannot vanish on $H$ or even have smaller $L^2$ norms than $e^{- C \lambda}$