Lower bounds on the Hausdorff measure of nodal sets

Abstract

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

Added detail to exposition (especially Proposition 1) and added references to recent results of Colding-Minicozzi and of Mangoubi. To appear in MRL