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In this note, we present the concavity of the minimal $L^2$ integrals relatedto multiplier ideals sheaves on Stein manifolds. As applications, we obtain anecessary condition for the concavity degenerating to linearity, acharacterization for 1-dimensional case, and a characterization for theequality in 1-dimensional optimal $L^{2}$ extension problem to hold.

Concavity of minimal $L^2$ integrals related to multipler ideal sheaves