The asymptotic Tian-Yau-Zelditch expansion on Riemann surfaces with Constant Curvature

Abstract

Let $M$ be a regular Riemann surface with a metric which has constant scalar curvature $ρ$. We give the asymptotic expansion of the sum of the square norm of the sections of the pluricanonical bundles $K_{M}^{m}$. That is, \[\sum_{i=0}^{d_{m}-1}\|S_{i}(x_{0})\|_{h_{m}}^{2} \sim m(1+\fracρ{2 m})+O(e^{-\frac{(\log m)^{2}}{8}}),\] where $\{S_{0},...,S_{d_{m}-1}\}$ is an orthonormal basis for $H^{0}(M, K_{M}^{m})$ for sufficiently large $m$.