On the lower bound of the inner radius of nodal domains

TL;DR

This paper investigates lower bounds on the inner radius of nodal domains of Laplacian eigenfunctions on Riemannian manifolds, improving known bounds in special cases and exploring the relation with eigenfunction distribution.

Contribution

It provides improved asymptotic lower bounds for the inner radius in real-analytic and negatively curved cases, and links eigenfunction distribution to nodal domain geometry.

Findings

Improved bounds for real-analytic manifolds.

Log-type improvements for negatively curved manifolds.

Relation established between eigenfunction distribution and nodal domain size.

Abstract

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $ \phi_\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 $ \phi_\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.