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 $\rho$. We give the asymptotic expansion of the sum of the square norm of the sections of the pluricanonical bundles $K_{M}^{m}$. That is, \[ \sum_{i=0}^{d_{m}-1} \|S_{i}(x_{0})\|_{h_{m}}^{2} \sim m(1+\frac{\rho}{2m}) + O\left( e^{-\frac{(\log m)^{2}}{8}} \right), \] where $\{S_{0},\cdots,S_{d_{m}-1}\}$ is an orthonormal basis for $H^{0}(M, K_{M}^{m})$ for sufficiently large $m$.