The sharp isoperimetric inequality for varifolds in codimension two

We prove a Michael-Simon inequality for a class of varifolds in Euclidean space that includes all integral varifolds with locally bounded first variation of arbitrary dimension and codimension. In codimension $2$, this inequality is sharp and implies the sharp isoperimetric inequality for varifolds, which attains equality precisely for the round disk. Our proof uses the optimal transportation approach of Brendle-Eichmair.