Lower bounds on the Hausdorff measure of nodal sets

Abstract

Let $\ncal_{\phi_{\lambda}}$ be the nodal hypersurface of a $\Delta$-eigenfunction $\phi_{\lambda}$ of eigenvalue $\lambda^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure: $\hcal^{n-1}(\ncal_{\phi_{\lambda}}) \geq C \lambda^{\frac74-\frac{3n}4} $. The best prior lower bound appears to be $e^{- C \lambda}$.