Nodal sets of eigenfunctions on Riemannian manifolds (Q1112407)

Let \(\Delta\) denote the Laplacian of a compact connected Riemannian manifold M. Suppose that F is a real eigenfunction of \(\Delta\) with eigenvalue \(\lambda\). It is proved that F vanishes to at most order \(c\sqrt{\lambda}\), for any point in M. The nodal set N of F is defined to be the set of points where F vanishes. If M is real analytic, upper and lower bounds are obtained for the n-1-dimensional Hausdorff measure of N. More specifically, \(c_ 1\sqrt{\lambda}\leq {\mathcal H}^{n-1}N\leq c_ 2\sqrt{\lambda}\).