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

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.