Curvature estimates for the metric associated with the Szegő kernel

Determine estimates for the Gaussian curvature κ_Szegő of the metric on a multiply connected planar domain defined by the Szegő kernel capacity function, for example ds = (2π·K_Szegő(z,z))|dz|, in analogy with the known bounds κ_electro ≤ −4 and κ_hydro ≤ −4 for the electrostatic and hydrodynamic metrics.

Background

The paper introduces several capacity functions tied to different kernels (electrostatic Bergman, hydrodynamic Bergman, and Szegő) and observes that these functions act as coefficients of natural conformal metrics. For the electrostatic and hydrodynamic metrics the authors recall known curvature bounds (≤ −4).

However, for the metric associated to the Szegő kernel the authors report no known curvature estimate. Establishing bounds for κ_Szegő would complete the parallel with the electrostatic and hydrodynamic cases and clarify the geometric strength of the Szegő-based metric.