On the Inner Radius of Nodal Domains

Abstract

Let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue \lambda. We give upper and lower bounds on the inner radius of the type C/\lambda^k. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman, and a Poincar\'e type inequality proved by Maz'ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.