Vol. 195, No. 1, 2000

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Riemannian manifolds admitting isometric immersions by their first eigenfunctions

Ahmad El Soufi and Saïd Ilias

Vol. 195 (2000), No. 1, 91–99
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 2 are the only λ1-minimal and the only critical ones.

Milestones
Received: 20 October 1998
Revised: 15 June 1999
Published: 1 September 2000
Authors
Ahmad El Soufi
Laboratoire de mathematiques et physique theorique
Universite de Tours
Parc de Grandmont
37200 Tours
France
Saïd Ilias
Laboratoire de mathematiques et physique theorique
Universite de Tours
Parc de Grandmont
37200 Tours
France