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In this article, we consider a modified version of minimal $L^2$ integrals onsublevel sets of plurisubharmonic functions related to modules at boundarypoints, and obtain a concavity property of the modified version. As anapplication, we give a characterization for the concavity degenerating tolinearity on open Riemann surfaces.

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