Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three

Abstract

Let ${\Delta }_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:\Delta u+\lambda u=0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate by showing that $H^1(\{u=0\})\le C{\lambda }^{3/4-\beta }$, $\beta \in (0,1/4)$. The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension $n=3$: $H^2(\{u=0\})\ge c{\lambda }^{\alpha }$, $\alpha \in (0,1/2)$. The positive constants $c,C$ depend on the manifold, $\alpha$ and $\beta$ are universal.

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