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

Abstract

Let $u$ be a harmonic function in the unit ball $B(0,1)\subset {\mathbb{R}}^n$, $n\ge 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $H^{n-1}(\{u=0\}\cap B)\ge c$. 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 ${\phi }_{\lambda }$ on $M$, which corresponds to the eigenvalue $\lambda$, the following inequality holds: $c\sqrt{\lambda }\le H^{n-1}(\{{\phi }_{\lambda }=0\})$.