Concavity property of minimal $L^2$ integrals with Lebesgue measurable gain III: open Riemann surfaces

In this article, we present a characterization of the concavity property of minimal $L^2$ integrals degenerating to linearity in the case of finite points on open Riemann surfaces. As an application, we give a characterization of the holding of equality in optimal jets $L^2$ extension problem from analytic subsets to open Riemann surfaces, which is a weighted jets version of Suita conjecture for analytic subsets.