  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.
