  In this article, we present characterizations of the concavity property of minimal $L^2$ integrals degenerating to linearity in the case of fibrations over open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets $L^2$ extension problem from fibers over analytic subsets to fibrations over open Riemann surfaces, which implies characterizations of the fibration versions of the equality parts of Suita conjecture and extended Suita conjecture. 
Concavity property of minimal $L^2$ integrals with Lebesgue measurable   gain V--fibrations over open Riemann surfaces