Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three
Alexander Logunov, Eugenia Malinnikova · arXiv (Cornell University) · 2016
Let $Δ_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:Δu + λu =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate by showing that $H^1(\{u=0 \})\le Cλ^{3/4-β}$, $β\in (0,1/4)$. The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension $n=3$: $H^2(\{u=0\})\ge cλ^α$, $α\in (0,1/2)$. The positive constants $c,C$ depend on the manifold, $α$ and $β$ are universal.