Boundary points, minimal $L^2$ integrals and concavity property III -- linearity on Riemann surfaces

TL;DR

This paper studies a modified minimal $L^2$ integral on Riemann surfaces, revealing a concavity property that degenerates to linearity, providing new insights into boundary behavior and integrals in complex analysis.

Contribution

It introduces a modified minimal $L^2$ integral framework and characterizes when the associated concavity becomes linear on Riemann surfaces.

Findings

Concavity of the modified $L^2$ integral is established.

Linear degeneracy of the concavity is characterized on Riemann surfaces.

Application to boundary point modules enhances understanding of complex boundary behavior.

Abstract

In this article, we consider a modified version of minimal $L^2$ integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points, and obtain a concavity property of the modified version. As an application, we give a characterization for the concavity degenerating to linearity on open Riemann surfaces.