Local Asymmetry and the Inner Radius of Nodal Domains

Let M be a closed Riemannian manifold of dimension n. Let φλ be an eigenfunction of the Laplace–Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {φλ > 0} ∩B is ≥ C|B|/λ , where B is any ball centered at a point of the nodal set. We apply this result to prove that each nodal domain contains a ball of radius ≥ C/λ. The results in this paper extend previous results of F. Nazarov, L. Polterovich, and M. Sodin, and of the author.