Sharp local Bernstein estimates for Laplace eigenfunctions on Riemannian manifolds

In this paper we focus on local growth properties of Laplace eigenfunctions on a compact Riemannian manifold. The principal theme is that a Laplace eigenfunction behaves locally as a polynomial function of degree proportional to the square root of the eigenvalue. In this direction, we notably prove sharp local Bernstein estimates, conjectured by Donnelly and Fefferman in 1990.