Lower bounds on the Hausdorff measure of nodal sets

Details Lower bounds on the Hausdorff measure of nodal sets Lower bounds on the Hausdorff measure of nodal sets

2010-09-18

arxiv.org/abs/1009.3573v3

Math. Res. Lett. 18 (2011), no. 1, 25-37

Mathematics

Analysis of PDEs

Added detail to exposition (especially Proposition 1) and added references to recent results of Colding-Minicozzi and of Mango

Scientific paper

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 $e^{- C \lambda}$.

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