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 $\varphi_\lambda$ on $M$, which corresponds to the eigenvalue $\lambda$ the following inequality holds: $c\sqrt{\lambda} \le H^{n-1}(\{\varphi_\lambda = 0\})$.