The inner radius of nodal domains in high dimensions

Abstract

We prove that every nodal domain of an eigenfunction of the Laplacian of eigenvalue $\lambda$ on a $d$-dimensional closed Riemannian manifold contains a ball of radius $c{\lambda }^{-1/2}(\log \lambda {)}^{-(d-2)/2}$. This ball is centered at a point at which the eigenfunction attains its maximum in absolute value within the nodal domain.

arXiv comment15 pages. v2: Presentation of proof of main theorem significantly simplified following Sasha Logunov's suggestion