Suita conjecture for a punctured torus

TL;DR

This paper investigates the relationship between the Bergman kernel and the fundamental metric on a punctured torus, extending Suita conjecture results to a broader class of Riemann surfaces through explicit potential construction.

Contribution

It explicitly constructs the Evans-Selberg potential for a punctured torus and discusses its asymptotics, generalizing Suita conjecture insights to parabolic Riemann surfaces.

Findings

Explicit Evans-Selberg potential constructed

Asymptotic behaviors analyzed

Generalization of Suita conjecture to parabolic surfaces

Abstract

For a once-punctured complex torus, we compare the Bergman kernel and the fundamental metric, by constructing explicitly the Evans-Selberg potential and discussing its asymptotic behaviors. This work aims to generalize the Suita type results to potential-theoretically parabolic Riemann surfaces.