Nodal intersections and Geometric Control

TL;DR

This paper generalizes results on the nodal points of eigenfunctions from plane domains to all real analytic Riemannian manifolds, providing geometric conditions for the eigenfunctions' restrictions and intersections with hypersurfaces.

Contribution

It extends previous work to higher dimensions and manifolds, establishing geometric control conditions for the 'goodness' of hypersurfaces related to eigenfunction behavior.

Findings

Lower bounds on eigenfunction restrictions to hypersurfaces.

Geometric conditions ensuring eigenfunctions do not vanish on hypersurfaces.

Partial answer to Bourgain-Rudnick's question on eigenfunction vanishing.

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 $e^{- C \lambda}$