On a Nonlinear Dirac Equation of Yamabe Type

We show a conformal spectral estimate for the Dirac operator on a non-conformally-flat Riemannian spin manifolds of dimension n ≥ 7. The estimate is a spinorial analogue to an estimate by Aubin which solved the Yamabe problem for the above manifolds. Using Hijazi’s inequality our estimate implies Aubin’s estimate. More exactly, let M be a compact manifold of dimension n ≥ 7 equipped with a Riemannian metric g and a spin structure σ. Assume that M is not conformally flat. Let λ+1 (g̃) be the smallest positive eigenvalue of the Dirac operator D on M with respect to a metric g̃ conformal to g. We define λ+min(M, g, σ) := inf g̃∈[g] λ + 1 (g̃)Vol(M, g̃) 1/n. In this article we show λ + min(M, g, σ) < λ + min(S ) = n 2 Vol(S) 1 n . Applying this inequality to a conformally invariant functional containing the Dirac operator, one can rule out that a minimizing sequence concentrates somewhere. We obtain applications to conformal spectral theory and to a nonlinear partial differential equation with a critical nonlinearity. MSC 2000: 53 A 30, 53C27 (Primary) 58 J 50, 58C40 (Secondary)