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On the lower bound of the inner radius of nodal domains

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