Title: Eigen functions with prescribed nodal sets.
Author(s): Alberto Enciso (CSIC-ICMAT) and Daniel Peralta-Salas (CSIC-ICMAT).
Source: Journal of Differential Geometry 101 (2015) 197-211.
Date of publication: September 16, 2015.
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? in a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is ?. 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.