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The first eigenvalue of Dirac and Laplace operators on surfaces

Jean-Francois Grosjean·Emmanuel Humbert
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Abstract

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

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Jean-Francois Grosjean, Emmanuel Humbert. 2006-09-18. The first eigenvalue of Dirac and Laplace operators on surfaces. https://arxiv.org/abs/math/0609493

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