A Spinorial Analogue of Aubin’s Inequality

Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric g̃ conformal to g, we denote by λ̃ the first positive eigenvalue of the Dirac operator on (M, g̃, σ). We show that inf g̃∈[g] λ̃ Vol(M, g̃) ≤ (n/2) Vol(S). This inequality is a spinorial analogue of Aubin’s inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case n ≥ 3 and in the case n = 2, kerD = {0}. Our proof also works in the remaining case n = 2, kerD 6= {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with 2λ̃2 ≤ μ̃, where μ̃ denotes the first positive eigenvalue of the Laplace operator. MSC 2000: 53 A 30, 53C27 (Primary) 58 J 50, 58C40 (Secondary)