I recently stumbled across a quote of Fang-Hua Lin that I have trouble understanding [1, page 42].



> 
> 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 $T$ is 'stationary' if the varifold $\lvert T \rvert$ obtained by forgetting orientations is stationary. If I am not mistaken, this should mean that $\partial T = 0$? However my impression is that a flat limit $T$ would be 'stationary' in this sense of the word.


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



How to interpret this quote of Lin?
I believe the following sequence demonstrates the failure of flat limits to be stationary. This would be consistent with the natural interpretation of the quote, meaning: a current $T$ is called *stationary* if the varifold $\lvert T \rvert$ is.


(A quick side remark before the construction: on second thought whether $\partial T = 0$ or not seems seems unrelated to the stationarity of $\lvert T \rvert$. For example, a triple junction has boundary as a current, but is stationary as a varifold. A slightly simpler variant of the example below would see the current $S$ replaced with a triple junction.)


That being said, the currents in the constructed sequence $(T\_n \mid n \in \mathbf{N})$ are one-dimensional cycles in the unit disc $D \subset \mathbf{R}^2$: $\partial T\_n = 0$ for all $n$. They converge weakly as currents to another cycle, say $T\_n \to T$ as $n \to \infty$. Most important: $\lvert T\_n \rvert$ is stationary for all $n$, but $\lvert T \rvert$ is not.


To construct the sequence, let $\{ v\_1,\dots,v\_6 \} \subset \partial D$ be unit vectors with
\begin{equation}
v\_1 + \cdots + v\_6 = 0,
\end{equation}
but which do not match up into antipodal pairs. For example
\begin{equation}
-v\_1 \not \in \{ v\_1,\dots,v\_6 \}.
\end{equation}


* Let $S \in I\_1(D)$ be the current supported in the union of the segments $\{ t v\_i \mid 0 \leq t \leq 1 \}$, oriented so that
$\partial S = 0.$
The associated varifold $\lvert S \rvert$ is stationary by construction.
* Let $L$ be the current supported in the segment $\{ tv\_1 \mid -1 \leq t \leq 1 \}$, which we orient in the *opposite* direction. In other words
\begin{equation}
\{ t v\_1 \mid 0 < t \leq 1 \} \cap \mathrm{spt} \, (S + L) = \emptyset.
\end{equation}
This too has $\partial L = 0$ and $\lvert L \rvert$ stationary.


The orientations are chosen so as to ensure that the current $T := S + L$ is not stationary; this is because $-v\_1 + \cdots + v\_6 = -2v\_1 \neq 0$.


Next we consider a sequence of positive angles $\theta\_n \to 0$. We use these angles to rotate $L$, forming a sequence of currents
\begin{equation}
R\_{\theta\_n \#} L \to L
\text{ as $n \to \infty$.}
\end{equation}
As long as $\theta\_n$ is small enough that $\mathrm{spt} \, R\_{\theta\_n \#} L \cap \mathrm{spt} \, S = \{ 0 \}$, the cycles $T\_n := S + R\_{\theta\_n \#} L$ are stationary. However
\begin{equation}
T\_n = S + R\_{\theta\_n \#}L \to S + L = T \text{ as $n \to \infty$}
\end{equation}
in the current topology, which was pointed out above is not stationary.


