The first eigenvalue of Dirac and Laplace operators on surfaces

Jean-Francois Grosjean (1) , Emmanuel Humbert (1)

1 IECN - Institut Élie Cartan de Nancy

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 $$\inf \frac{\lambda_1(\tilde{g})^2 }{\mu_1(\tilde{g})} \leqslant \frac{1}{2}.$$ where the infimum is taken over the metrics $\tilde{g}$ conformal to $g$. This answer a question asked by Agricola, Ammann and Friedrich

Mots clés: Surface; spectrum; conformal geometry

Domaines: Géométrie différentielle [math.DG]

Jean-Francois Grosjean, Emmanuel Humbert. The first eigenvalue of Dirac and Laplace operators on surfaces. 2006. ⟨hal-00095771⟩