Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Alexander Logunov¹

¹ School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel<br/> Chebyshev Laboratory, St. Petersburg State University, Saint Petersburg, Russia <br/>Institute for Advanced Study, Princeton, NJ, USA

Annals of mathematics, Tome 187 (2018) no. 1, pp. 221-239

Résumé

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $\varphi_\lambda: \Delta_M \varphi_\lambda + \lambda \varphi_\lambda = 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $\lambda$. We show that $$H^{n-1}(\{ \varphi_\lambda=0\}) \leq C \lambda^{\alpha},$$ where $\alpha>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index. % We obtain partial positive answers to the question: Is the frequency additive in some sense?

Détail

DOI : 10.4007/annals.2018.187.1.4