I have stumbled across a quote of Fang-Hua Lin [1] that I am having trouble understanding. 
> It is a well-known fact that a weakly converging sequence of stationary integral currents may have a limit which is not a stationary current.

**Question.** How should I interpret this quote? What does Lin mean by a 'stationary current', and which sequence demonstrates this 'well-known fact'?

My initial guess would be that an integral current $T$ (in the unit ball $B \subset \mathbf{R}^{n+k}$ say) is *stationary* if the varifold $\lvert T \rvert$ obtained by forgetting orientations is stationary in $B$. If I am not mistaken, this should mean that $\partial T = 0$?

But then if $(T_j \mid j \in \mathbf{N})$ is a sequence of currents in $B$ with bounded mass and $\partial T_j = 0$, then a subsequence will converge to an integral cycle $T$ in the flat topology. Of course it may happen that $T_j \to T$ but $\lvert T_j \rvert \not \to \lvert T \rvert$ because of mass cancellation. However, I cannot think of an example where $\lvert T \rvert$ would not be stationary.


[1] F.-H. Lin. Mapping problems, fundamental groups and defect measures. Acta Math. Sin. 15 (1999), 25-52.

[2] W. Allard. First variation of a varifold. Annals of Math. 95 (1972), 417-491.