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 suficient 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.

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