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

Let u be a harmonic function in the unit ball B(0; 1) ⊂ ℝⁿ, n ≥ 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) (n-ary intersection) B) ≥ c. We prove Nadirashvili's conjecture as well as its counterpart on C∞-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjec- ture. Namely, we show that for any compact C∞-smooth Riemannian manifold M (without boundary) of dimension n, there exists c > 0 such that for any Laplace eigenfunction ψλ on M, which corresponds to the eigenvalue λ, the following inequality holds: c λ ≤ H^n-1((ψλ=0))).