Local Asymmetry and the Inner Radius of Nodal Domains

Abstract

Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/λ^n. We apply this result to prove that each nodal domain contains a ball of radius > C/λ^n.

12 pages, 1 figure; minor corrections; to appear in Comm. PDEs

Cited by in corpus (6)

• Lower bounds for nodal sets of eigenfunctions
• Nodal Geometry, Heat Diffusion and Brownian Motion
• A Remark on Recent Lower Bounds for Nodal Sets
• Nodal set of monochromatic waves satisfying the Random Wave Model
• Density of zero sets for sums of eigenfunctions
• Hausdorff measure bounds for nodal sets of Steklov eigenfunctions