A Local Courant Theorem on Real Analytic Manifolds
Dan Mangoubi · arXiv (Cornell University) · 2008
Let M be a closed real analytic Riemannian manifold. We estimate from below the volume of a nodal domain component in an arbitrary ball, provided that this component enters the ball deeply enough. We also improve the existing estimates in the smooth case. 1 Introduction and Main Results Let (M, g) be a closed C ∞-Riemannian Manifold of dimension n. Let ∆ be the Laplace–Beltrami operator on M. We consider the eigenvalue equation ∆ϕλ = λϕλ (1.1) The null set {ϕλ = 0} is called the λ-nodal set and any connected component