Nodal sets for sums of eigenfunctions on Riemannian manifolds

Abstract

Quantitative versions of unique continuation are proved for finite sums of eigenfunctions of the Laplacian on compact Riemannian manifolds. The results include a lower bound for the order of vanishing, a growth estimate for the supremum on compact balls, and a gradient bound. For real analytic metrics, an upper bound for the Hausdorff measure of the zero set is derived.