The first eigenvalue of Dirac and Laplace operators on surfaces

Abstract

Let $(M,g,\sigma )$ be a compact Riemmannian surface equipped with a spin structure $\sigma$. For any metric $\tilde{g}$ on $M$, we denote by $\mu \_1(\tilde{g})$ (resp. $\lambda \_1(\tilde{g})$) the first positive eigenvalue of the Laplacian (resp. the Dirac operator) with respect to the metric $\tilde{g}$. In this paper, we show that $$\inf \frac{\lambda \_1(\tilde{g}{)}^2}{\mu \_1(\tilde{g})}⩽1/2.$$ where the infimum is taken over the metrics $\tilde{g}$ conformal to $g$. This answer a question asked by Agricola, Ammann and Friedrich