Federer-Fleming and Allard are somewhat complementary. Integral currents are a special kind of distribution valued forms, while varifolds are a special kind of measures on the real Grassmannian bundle $Gr(TX,m)$ over $X$ whose fibres parametrize $m$-dimensional planes in the tangent spaces of $X$. One key advantage of currents is that they know about orientations, while varifolds do not. The integral current $L$ recovers the underlying rectifiable subset $\text{supp}(L)$ with multiplicity, so can be canonically associated with a varifold $L^{var}$. On the other hand, the natural topology on varifolds (i.e. the topology as measures on $Gr(TX,m)$) remembers tangent plane information, which can be lost under the flat norm convergence of integral currents. Morever, assuming all the varifolds in the sequence are contained in a bounded region, then the total volume mass converges under varifold convergence, but not necessarily so under flat norm convergence. The intuition is that morally the varifold topology detects one more derivative than the flat norm topology. This explains why Allard requires some integral control on the mean curvature, while Federer-Fleming does not.

We shall later use the informal terminology of ‘varifold/current topology’ to refer to convergence simultaneously in the varifold topology and the flat norm topology on integral currents.