Concavity property of minimal $L^2$ integrals with Lebesgue measurable gain III: open Riemann surfaces

TL;DR

This paper investigates the concavity of minimal $L^2$ integrals on open Riemann surfaces, providing characterizations of when this property degenerates to linearity and applying it to optimal extension problems related to the Suita conjecture.

Contribution

It offers a new characterization of the concavity property of minimal $L^2$ integrals and applies it to the optimal jets $L^2$ extension problem on open Riemann surfaces.

Findings

Characterization of when minimal $L^2$ integrals degenerate to linearity.

Conditions for equality in the optimal jets $L^2$ extension problem.

Weighted jets version of the Suita conjecture for analytic subsets.

Abstract

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.