  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.
