Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure. (English) Zbl 1384.58020

Let \(\mathcal{M}=(M,g)\) be a compact smooth Riemannian manifold of dimension \(n\geq3\). Let \(\Delta\) be the scalar Laplacian. Suppose that \(\phi_\lambda\) is a smooth eigenfunction of the Laplacian corresponding to the eigenvalue \(\lambda\), i.e. \(\Delta\phi_\lambda+\lambda\phi_\lambda=0\). Let \(H^{n-1}\) denote the \((n-1)\) dimensional Hausdorff measure. Let \(N(\phi_\lambda):=\{x:\phi_\lambda(x)=0\}\) be the nodal set. The author shows that there exists a constant \(\alpha(n)>\frac12\) and a constant \(C(\mathcal{M})>0\) so that \(H^{n-1}(N(\phi_\lambda))\leq C(\mathcal{M})\lambda^{\alpha(n)}\). This estimate is derived from an estimate bounding the volume of the nodal set of a harmonic function in terms of the frequency function.
The first section contains a historical summary putting the problem in context. The second section treats the simplex lemma. The third section discusses the propagation of smallness of Cauchy data. The fourth section provides the hyperplane lemma. The fifth section gives the number of cubes with a big doubling index. The sixth section provides upper estimates of the volume of the nodal set. The final section contains a number of auxiliary lemmas.


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P05 General topics in linear spectral theory for PDEs
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