We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
"Regularity theory for $2$-dimensional almost minimal currents III: Blowup." J. Differential Geom. 116 (1) 125 - 185, September 2020. https://doi.org/10.4310/jdg/1599271254