Immersions minimales, première valeur propre du laplacien et volume conforme

Riemannian manifolds admitting isometric immersions by their first eigenfunctions

“…Using results established by us in [4] about λ 1 -minimal metrics we deduce that (Corollary 1.1), if g is an extremal metric of the λ 1 functional then:…”

“…As (M, g) is λ 1 -minimal non homothetic to S n then any conformal diffeomorphism of (M, g) is an isometry (cf. [4]). It follows that any isometry of (M, g 0 ) is also an isometry of (M, g).…”

“…In [4] we showed that λ 1 -minimal metrics satisfy certain remarkable conformal properties. Theorem 1.1 tells us that all these properties are still true for extremal metrics: Corollary 1.1.…”

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

“…Note that conditions of this type concerning the "unconstrained" critical metrics were obtained by Nadirashvili [6] in dimension 2 and by the authors [7] in all dimensions. The necessary condition means that there exists a harmonic map f from (M, g) to a unit sphere with energy density |df | 2 = λ 1 (g) (see [5]).…”

“…It is known that if n ≥ 3, then the functional λ 1 is unbounded (see [3]). However, for any metric g on M , the restriction of λ 1 to the conformal class C(g) = {φg ; φ > 0 and V (φg) = V (g)} of g is bounded (see [6] and [10]). Here V (g) denotes the Riemannian volume of g. This paper consists of two sections.…”