Capacity estimates and improved lower bounds for the inner radius of nodal domains

Abstract

We show that for every closed, smooth manifold $(M,g)$ of dimension $d$, there exists $c(g)$ such that any nodal domain ${\Omega }_{\lambda }$ of a Laplace eigenfunction with eigenvalue $\lambda$ contains a geodesic ball of radius at least $c(g){\lambda }^{-1/2}\log \log (\lambda {)}^{-1/2}$ if $d=3$ and $c(g){\lambda }^{-1/2}\log (\lambda {)}^{-\frac{d-3}{2}}$ if $d>3$. This ball is centered at any point at which the eigenfunction attains its maximum in absolute value within the nodal domain. Furthermore, we show that for any $d\ge 3$, there exist sequences of $\lambda$-nodal domains on ${\mathbb{T}}^d$ whose inner radius is of order $o({\lambda }^{-1/2})$.