Un problème de type Yamabe sur les variétés spinorielles compactes

Abstract

Let ( M, g, σ) be a compact spin manifold of dimension n⩾2. Let λ 1 +( g ̃ ) be the smallest positive eigenvalue of the Dirac operator in the metric g ̃ ∈[g] conformal to g. We then define λ min +(M,[g],σ)= inf g ̃ ∈[g] λ 1 +( g ̃ ) Vol(M, g ̃ ) 1/n . We show that 0<λ min +(M,[g],σ)⩽λ min +( S n) . We find sufficient conditions for which we obtain strict inequality λ min +(M,[g],σ)<λ min +( S n) . This strict inequality has applications to conformal spin geometry. To cite this article: B. Ammann et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).