Riemannian manifolds admitting isometric immersions by their first eigenfunctions

Ahmad El Soufi and Saïd Ilias

Vol. 195 (2000), No. 1, 91–99

DOI: 10.2140/pjm.2000.195.91

Abstract

Given a compact manifold M, we prove that every critical Riemannian metric g for the functional “first eigenvalue of the Laplacian” is λ₁-minimal (i.e., (M,g) can be immersed isometrically in a sphere by its first eigenfunctions) and give a sufficient condition for a λ₁-minimal metric to be critical. In the second part, we consider the case where M is the 2-dimensional torus and prove that the flat metrics corresponding to square and equilateral lattices of ℝ² are the only λ₁-minimal and the only critical ones.

Milestones

Received: 20 October 1998

Revised: 15 June 1999

Published: 1 September 2000