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

TL;DR

This paper proves Nadirashvili's conjecture on the lower bound of nodal set measures for harmonic functions and establishes the lower bound in Yau's conjecture for Laplace eigenfunctions on smooth manifolds.

Contribution

It confirms Nadirashvili's conjecture and extends the lower bound result to Laplace eigenfunctions, advancing understanding of nodal sets in geometric analysis.

Findings

Proof of Nadirashvili's conjecture on harmonic functions.

Establishment of the lower bound in Yau's conjecture.

Extension of results to smooth Riemannian manifolds.

Abstract

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: $c \sqrt \lambda \leq H^{n-1}(\{\varphi_\lambda =0\})$.