Eigen functions with prescribed nodal sets

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.

Link: https://projecteuclid.org/download/pdf_1/euclid.jdg/1442364650