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

TL;DR

This paper characterizes when minimal $L^2$ integrals on product Riemann surfaces are linear, leading to new insights into equality conditions in extension problems and conjectures like Suita and Ohsawa's for these surfaces.

Contribution

It provides new characterizations of the concavity and linearity of minimal $L^2$ integrals on product Riemann surfaces, extending previous results to product settings.

Findings

Characterization of linearity conditions for minimal $L^2$ integrals on product Riemann surfaces.

New criteria for equality in optimal jets $L^2$ extension problems on product surfaces.

Implications for the product versions of the Suita and Ohsawa conjectures.

Abstract

In this article, we present characterizations of the concavity property of minimal $L^2$ integrals degenerating to linearity in the case of products of analytic subsets on products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets $L^2$ extension problem from products of analytic subsets to products of open Riemann surfaces, which implies characterizations of the product versions of the equality parts of Suita conjecture and extended Suita conjecture, and the equality holding of a conjecture of Ohsawa for products of open Riemann surfaces.