Riemannian manifolds admitting isometric immersions by their first eigenfunctions

Abstract

Given a compact manifold M, we prove that every critical Riemannian metric g for the functional "first eigenvalue of the Laplacian" is λ 1 -minimal (i.e., (M, g) can be immersed isometrically in a sphere by its first eigenfunctions) and give a sufficient condition for a λ 1 -minimal metric to be critical.In the second part, we consider the case where M is the 2dimensional torus and prove that the flat metrics corresponding to square and equilateral lattices of R 2 are the only λ 1minimal and the only critical ones.