estimate on the Hausdorff measure $$\begin{aligned}
\label{e:haus_est_nod}
{\mathcal{H}}^{n-2}(Z(u)\cap B_{r_0}(0))\leq C^{\Lambda^2}\, .\end{aligned}$$

#### Applications to Eigenvalue Equation on Manifolds:

Let us now assume we are working on a compact Riemannian manifold $(M^n,g)$ with at least Lipschitz metric $g$. In this context we are most interested in studying the Laplace beltrami operator $\Delta u \equiv \text{div} \nabla u$, though the results hold equally well for other second order operators. It is well understood that the eigenvalues $0=\lambda_0<\lambda_1\leq \lambda_2\leq \cdots$ of $-\Delta$ are discrete with $\lambda_j\to \infty$. As an application of Theorems \[t:main\_critical\] and \[t:main\_nodal\], as well as the doubling estimate of [@DonFef] we have the following:

For a compact lipschitz Riemannian manifold $(M^n,g)$ there exists a constant $C(g)$ such that if $u$ solves the eigenfunction equation $-\Delta u = \lambda u$ then we have the Minkowski estimates $$\begin{aligned}
&{\operatorname{Vol}}(B_r({\mathcal{S}}(u))) \leq {\operatorname{Vol}}(B_r({\mathcal{S}}_r(u))) \leq C^{\lambda}\,r^2\, ,\notag\\
&{\operatorname{Vol}}(B_r(Z(u))) \leq {\operatorname{Vol}}(B_r(Z_r(u))) \leq \lambda^{C\sqrt{\lambda}}\,r\, .\end{aligned}$$ In particular, we have the much weaker estimate on the Hausdorff measure $$\begin{aligned}
&{\mathcal{H}}^{n-2}({\mathcal{S}}(u))\leq C^{\lambda}\, ,\notag\\
&{\mathcal{H}}^{n-1}(Z(u))\leq \lambda^{C\sqrt{\lambda}}\, .\end{aligned}$$

Preliminaries and Outline of Proof
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In this section we concentrate on introducing the correct terminology for the paper, as well as giving an outline for the proof of the mains Theorems. To 