Nodal sets of laplace eigenfunctions: Estimates of the hausdorff measure in dimensions two and three

Abstract

Let ΔM be the Laplace operator on a compact n-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u: ΔMu+λu = 0. In dimension n = 2 we refine the Donnelly–Fefferman estimate by showing that H1({u = 0}) ≤Cλ3/4−β for some β∈ (0, 1/4). The proof employs the Donnelly–Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension n = 3: H2({u = 0}) ≥ cλα for some α∈ (0,1/2). The positive constants c, C depend on the manifold, α and β are universal.