A spinorial analogue of Aubin's inequality

Abstract

Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\ge 2$. For any metric $\tilde{g}$ conformal to $g$, we denote by $\tilde{\lambda }$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. 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 $n\ge 3$ and in the case $n=2$, $\ker D=\{0\}$. Our proof also works in the remaining case $n=2$, $\ker D\ne \{0\}$. With the same method we also prove that any conformal class on a Riemann surface contains a metric with $2{\tilde{\lambda }}^2\le \tilde{\mu }$, where $\tilde{\mu }$ denotes the first positive eigenvalue of the Laplace operator.

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