Mathematical Research Letters

Volume 18 (2011)

Number 1

Lower bounds on the Hausdorff measure of nodal sets

Pages: 25 – 37



Christopher D. Sogge (Johns Hopkins University, Baltimore, MD 21218, USA)

Steve Zelditch (Northwestern University, Evanston IL, 60208-2730, USA)


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 that $\hcal^{n-1}(\ncal_{\phi_{\lambda}}) \geq C \lambda^{\frac74-\frac{3n}4} $. %on the surface measure of its nodal set. The best previous lower bound was $e^{- C \lambda}$.

Published 28 February 2011