A Yamabe type problem on compact spin manifolds

Let (M, g, sigma) be a compact spin manifold of dimension n greater than or equal to 2. Let lambda(1)(+)((g) over tilde) be the smallest positive eigenvalue of the Dirac operator in the metric (g) over tilde is an element of [g] conformal to g. We then define lambda(min)(+)(M, [g], sigma) = inf((g) over tilde is an element of[g]) lambda(1)(+)((g) over tilde) Vol(M, (g) over tilde)(1/n). We show that 0 < lambda(min)(+)(m, [g], sigma) less than or equal to lambda(min)(+)(S-n). We find sufficient conditions for which we obtain strict inequality lambda(min)(+)(M, [g], sigma) < lambda(min)(+)(S-n). This strict inequality has applications to conformal spin geometry. (C) 2004 Academie des sciences. Publie par Elsevier SAS. Tous droits reserves.