Concavity property of minimal $L^2$ integrals with Lebesgue measurable gain VI: fibrations over products of open Riemann surfaces

TL;DR

This paper investigates the concavity of minimal $L^2$ integrals in fibrations over products of open Riemann surfaces, providing characterizations of equality conditions related to the Suita conjecture and its extension.

Contribution

It offers new characterizations of when the minimal $L^2$ integrals degenerate to linearity in fibrations over open Riemann surfaces, linking to the Suita conjecture.

Findings

Characterizations of concavity and linearity of minimal $L^2$ integrals.

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

Implications for the Suita and extended Suita conjectures.

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 products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets $L^2$ extension problem from fibers over products of analytic subsets to fibrations over products of open Riemann surfaces, which implies characterizations of the equality parts of Suita conjecture and extended Suita conjecture for fibrations over products of open Riemann surfaces.