On the Inner Radius of Nodal Domains.

Let $M$ be a closed Riemannian manifold of dimension $n$.
Let $\phi$ be an eigenfunction of the Laplacian on $M$ with eigenvalue lamda.
A nodal domain is a connected component of the set $\phi <> 0$.

We discuss the asymptotic geometry of nodal domains on $M$.
We prove that the inner radius $R$ of a nodal domain is bounded by

$$C_1 / \sqrt{\lambda} > R > C_2/\lambda^{(n-1)/2} .$$

In dimension two we have a sharp bound.
One ingredient of our proof is the estimation of the volume
of positivity of a harmonic function $u$ in the unit ball, with $u(0)>0$, in terms of its growth.