Abstract. Let u be a harmonic function in the unit ball B(0, 1) ⊂ R n , 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} ∩ 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 conjecture. 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 ϕ λ on M , which corresponds to the eigenvalue λ, the following inequality holds: c √ λ ≤ H n−1 ({ϕ λ = 0}).

Abstract. Let M be a compact C ∞ -smooth Riemannian manifold of dimension n, n ≥ 3, and let ϕ λ : ∆M ϕ λ + λϕ λ = 0 denote the Laplace eigenfunction on M corresponding to the eigenvalue λ. We show that

Let ∆M be the Laplace operator on a compact n-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u : ∆u + λu = 0. In dimension n = 2 we refine the Donnelly-Fefferman estimate by showing that H 1 ({u = 0}) ≤ Cλ 3/4−β , β ∈ (0, 1/4). The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension n = 3: H 2 ({u = 0}) ≥ cλ α , α ∈ (0, 1/2). The positive constants c, C depend on the manifold, α and β are universal.