  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.
