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

TL;DR

This paper derives the asymptotic expansion of the sum of squared norms of sections of pluricanonical bundles on Riemann surfaces with constant curvature, revealing detailed growth behavior as the tensor power increases.

Contribution

It provides the first explicit asymptotic expansion for the sum of squared norms of pluricanonical sections on such surfaces, including an exponentially small error term.

Findings

Asymptotic expansion of sum of squared norms derived

Explicit error term with exponential decay established

Results apply to Riemann surfaces with constant scalar 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, \[\sum_{i=0}^{d_{m}-1}\|S_{i}(x_{0})\|_{h_{m}}^{2} \sim m(1+\frac{\rho}{2 m})+O(e^{-\frac{(\log m)^{2}}{8}}),\] where $\{S_{0},...,S_{d_{m}-1}\}$ is an orthonormal basis for $H^{0}(M, K_{M}^{m})$ for sufficiently large $m$.