A spinorial analogue of Aubin’s inequality

Abstract

Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric \(\tilde g\) conformal to g, we denote by \(\tilde\lambda\) the first positive eigenvalue of the Dirac operator on \((M, \tilde g, \sigma)\) . We show that$${\rm inf}_{\tilde{g} \in [g]} \tilde\lambda {\rm Vol}(M,\tilde g)^{1/n} \leq (n/2) {\rm Vol}(S^n)^{1/n}.$$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, ker D = {0}. Our proof also works in the remaining case n = 2, ker D ≠ {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with \(2\tilde\lambda^2 \leq \tilde\mu\) , where \(\tilde\mu\) denotes the first positive eigenvalue of the Laplace operator.