Concavity property of minimal $L^{2}$ integrals with Lebesgue measurable gain II

TL;DR

This paper investigates the concavity of minimal $L^{2}$ integrals associated with multiplier ideal sheaves on weakly pseudoconvex Kähler manifolds, providing conditions for linearity and characterizations in $L^2$ extension problems.

Contribution

It establishes a concavity property of minimal $L^{2}$ integrals with Lebesgue measurable gain and characterizes cases of linearity and equality in extension problems.

Findings

Concavity property of minimal $L^{2}$ integrals proven.

Necessary condition for concavity degenerating to linearity.

Characterization of equality cases in $L^2$ extension on Riemann surfaces.

Abstract

In this article, we present a concavity property of the minimal $L^{2}$ integrals related to multiplier ideal sheaves with Lebesgue measurable gain on weakly pseudoconvex K\"ahler manifolds. As applications, we give a necessary condition for the concavity degenerating to linearity, and a characterization for the holding of the equality in optimal jets $L^2$ extension problem on open Riemann surfaces.