Equality in Suita's conjecture and metrics of constant Gaussian curvature

TL;DR

This paper offers a new proof of the equality condition in Suita's conjecture for open Riemann surfaces, linking the Bergman kernel, capacity, and geometric properties without relying on the $L^2$ extension theorem.

Contribution

It provides a novel proof avoiding the $L^2$ extension theorem, using Maitani and Yamaguchi's variation formula, and characterizes surfaces via constant Gaussian curvature of the Bergman kernel.

Findings

Characterization of Riemann surfaces with constant Gaussian curvature

Relation between Bergman kernel and disc quotients

Results on planar domains without Bergman-completeness

Abstract

Without using the $L^2$ extension theorem, we provide a new proof of the equality part in Suita's conjecture, which states that for any open Riemann surface admitting a Green's function, the Bergman kernel and the logarithmic capacity coincide at one point if and only if the surface is biholomorphic to a disc possibly less a relatively closed polar set. In comparison with Guan and Zhou's proof, our proof essentially depends on Maitani and Yamaguchi's variation formula for the Bergman kernel, and we explore the harmonicity in such variations. As applications, we characterize the above surface by the constant Gaussian curvature property of the Bergman kernel or metric, and also find the relations with disc quotients. Additionally, we obtain results on planar domains without the Bergman-completeness assumption.