2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132407Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$. For any metric $\tilde g$ conformal to $g$, we denote by $\tildeλ$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show that $$\inf_{\tilde{g} \in [g]} \tildeλ\Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$ 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 $n \geq 3$ and in the case $n = 2$, $\ker D=\{0\}$. Our proof also works in the remaining case $n=2$, $\ker D\neq \{0\}$. With the same method we also prove that any conformal class on a Riemann surface contains a metric with $2\tildeλ^2\leq \tildeμ$, where $\tildeμ$ denotes the first positive eigenvalue of the Laplace operator.Title changed, introduction modified, main result has changed, applications addedDifferential GeometryAnalysis of PDEsSpectral Theory53 A 30, 53C27 (Primary) 58 J 50, 58C40 (Secondary)text