Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Let M be a compact C-infinity-smooth Riemannian manifold of dimension n, n >= 3, and let phi lambda : Delta M phi lambda + lambda phi lambda = 0 denote the Laplace eigenfunction on M corresponding to the eigenvalue lambda. We show that Hn-1 ({phi lambda = 0}) <= C lambda(alpha), where alpha > 1/2 is a constant, which depends on n only, and C > 0 depends on M. This result is a consequence of our study of zero sets of harmonic functions on C-infinity-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.