  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.
