Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture

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. 241-262

Résumé

Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $$H^{n-1}(\{u=0 \} \cap B) \geq c.$$ We prove Nadirashvili’s conjecture as well as its counterpart on $C^\infty$-smooth Riemannian manifolds. The latter yields the lower bound in Yau’s conjecture. Namely, we show that for any compact $C^\infty$-smooth Riemannian manifold $M$ (without boundary) of dimension $n$, there exists $c>0$ such that for any Laplace eigenfunction $\varphi_\lambda$ on $M$, which corresponds to the eigenvalue $\lambda$, the following inequality holds: $c \sqrt \lambda \leq H^{n-1}(\{\varphi_\lambda =0\})$.

Détail

DOI : 10.4007/annals.2018.187.1.5