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Nodal intersections and geometric control. (English) Zbl 1458.35289

Summary: We prove that the number of nodal points on an \(\mathcal{S} - good\) real analytic curve \(\mathcal{C}\) of a sequence \(\mathcal{S}\) of Laplace eigenfunctions \(\varphi_j\) of eigenvalue \(-\lambda^2_j\) of a real analytic Riemannian manifold \((M, g)\) is bounded above by \(A_{g, \mathcal{C}} \lambda_j\). Moreover, we prove that the codimension-two Hausdorff measure \(\mathcal{H}^{m-2} (\mathcal{N}_{\varphi \lambda} \cap H)\) of nodal intersections with a connected, irreducible real analytic hypersurface \(H \subset M\) is \(\leq A_{g, H} \lambda_j\). The \(\mathcal{S} \)-goodness condition is that the sequence of normalized logarithms \(\frac{1}{\lambda_j} \operatorname{log} {\vert \varphi_j \vert}^2\) does not tend to \(-\infty\) uniformly on \(\mathcal{C} \), resp. \(H\). We further show that a hypersurface satisfying a geometric control condition is \(\mathcal{S} \)-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.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P05 General topics in linear spectral theory for PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

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