Sur la th\'eorie spectrale des m\'etriques int\'egrables sur une surface de Riemann compacte

Abstract

We continue the study of the spectral theory associated to integrable metrics, started in our previous paper arXiv:1301.1793 [math.SP]. We introduce the notion of 1-integrable metric on line-bundles on a compact Riemann surface. We extend the spectral theory of generalized Laplacians to line-bundles equipped with 1-integrable metrics. As an application, we recover the following identity: [\zeta'_{\Delta_{\bar{\mathcal{O}(m)}_\infty}}(0)=T_g\bigl((\p^1,\omega_\infty); \bar{\mathcal{O}(m)}_\infty \bigr),] obtained using direct computations in arXiv:1301.1792 [math.NT].