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, \[\sum_{i=0}^{d_{m}-1}\|S_{i}(x_{0})\|_{h_{m}}^{2} \sim m(1+\fracρ{2 m})+O(e^{-\frac{(\log m)^{2}}{8}}),\] where $\{S_0,...,S_{d_m-1}\}$ is an orthonormal basis for $H^0(M,K_M^m)$ for sufficiently large $m$.

9 pages, final version, to appear in Taiwanese J. Math