Eigenfunctions with prescribed nodal sets

Abstract

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface $\Sigma$ in a compact $n$-manifold $M$, there is a Riemannian metric on $M$ such that the nodal set of its first nontrivial eigenfunction is $\Sigma$. We present a number of variations on this result, which enable us to show, in particular, that the first nontrivial eigenfunction can have as many non-degenerate critical points as one wishes.