On the lower bound of the inner radius of nodal domains

2016/07/13 by Georgiev, Bogdan
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1607.03816

Abstract

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 (\citeMan1). Furthermore, using recent results of Hezari (\citeHezari, \citeHezari2) 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 L² 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.

Related

• Some remarks on nodal geometry in the smooth setting
• Inner radius of nodal domains of quantum ergodic eigenfunctions
• The inner radius of nodal domains in high dimensions
• The size of the nodal sets for the eigenfunctions of the smooth laplacian
• Arithmetic, zeros, and nodal domains on the sphere