Concavity property of minimal $L^2$ integrals with Lebesgue measurable gain V--fibrations over open Riemann surfaces

TL;DR

This paper investigates the concavity of minimal L^2 integrals in fibrations over open Riemann surfaces, providing characterizations of when this property degenerates to linearity and applications to optimal extension problems.

Contribution

It offers new characterizations of the concavity and linearity of minimal L^2 integrals in fibrations over open Riemann surfaces, linking to conjectures in complex analysis.

Findings

Characterization of concavity degenerating to linearity in minimal L^2 integrals.

Conditions for equality in optimal jets L^2 extension problems.

Implications for the fibration versions of the Suita conjecture.

Abstract

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.