On the lower bound of the inner radius of nodal domains

Abstract

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions ${\varphi }_{\lambda }$ on a closed Riemannian manifold $(M,g)$. First, in the real-analytic case we present an improvement of the currently best known bounds, due to Mangoubi (\cite{Man1}). Furthermore, using recent results of Hezari (\cite{Hezari}, \cite{Hezari2}) we obtain $\log$-type improvements in the case of negative curvature and improved bounds for $(M,g)$ possessing an ergodic geodesic flow. Second, we discuss the relation between the distribution of the $L^2$ norm of an eigenfunction ${\varphi }_{\lambda }$ and the inner radius of the corresponding nodal domains. In the spirit of the works of Colding-Minicozzi and Jakobson-Mangoubi, we consider a covering of good cubes and show that, if a nodal domain is sufficiently well covered by good cubes, then its inner radius is large.

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