Concavity property of minimal $L^{2}$ integrals with Lebesgue measurable gain VII -- Negligible weights

TL;DR

This paper characterizes the concavity of minimal $L^2$ integrals with negligible weights on fibrations over Riemann surfaces, providing insights into when these integrals degenerate to linearity and applications to optimal extension problems.

Contribution

It offers new characterizations of the concavity and linearity conditions for minimal $L^2$ integrals with negligible weights on complex fibrations, advancing understanding of extension problems.

Findings

Concavity degenerates to linearity under specific conditions.

Characterizations of equality in optimal $L^2$ extension problems.

Applications to fibrations over Riemann surfaces and their products.

Abstract

In this article, we present characterizations of the concavity property of minimal $L^2$ integrals with negligible weights degenerating to linearity on the fibrations over open Riemann surfaces and the fibrations over products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets $L^2$ extension problem with negligible weights on the fibrations over open Riemann surfaces and the fibrations over products of open Riemann surfaces.