Least area bounded by multiple of curves In this paper SOME EXTREMAL QUESTIONS FOR SIMPLICIAL COMPLEXES, the author discussed about minimal area of bounded by multiples of a curve. Say we have a (well-behaved) curve $\Gamma$, the solution to the minimal area problem is $S$. Now e consider $2\Gamma$, he solution to the minimal area problem is $S'$. But $area(S')<2area(S)$.
My question is what does $2\Gamma$ mean? Does it mean a similiar transformation in the coordinate multiplying by 2? If it is this case, then I cannot really imagine this is true.
 A: As @alesia points out $2\Gamma$ means to take the curve ``with multiplicity two" (this is usually understood to be in the context of currents or flat chains -- for an oriented curve you can think about it as tracing out the curve twice).  The reason this can give a lower area for the minimizer is that the space of competitors for $2\Gamma$ contains two times all competitors for $\Gamma$ but may also contain new surfaces.
I'll try to illustrate this in (relatively) non-technical manner:
If $M$ is a Mobius band in $\mathbb{R}^3$ and $\Gamma=\partial M$, then it is the case that $2\Gamma=\partial \tilde{M}$ where $\tilde{M}$ is the orientation double cover of $M$.  If it is the case that $M$ is the least area surface bounded by $\Gamma$ (among all orientable and non-orientable surfaces), then one should expect that the least area of orientable surface bounded by $\Gamma$, $N$, (which exists by appealing to geometric measure theory results) can satisfy $|N|>|M|$ (here $|N|$ and $|M|$ are the areas of $N$ and $M$, respectively).  However, the least area orientable surface bounded by $2\Gamma$ should have area at most $2|M|$ (since $\tilde{M}$ is a valid competitor) and so one has
$$|N'|\leq |\tilde{M}|= 2|M|<2|N|$$
where $N'$ is the least area orientable surface spanning $2\Gamma$.
EDIT: As pointed out in the comments the above ``example" doesn't work as $\tilde{M}$ can't have the claimed properties.  The general idea is still correct and it seems can be made rigorous for a curve in $\mathbb{R}^4$.
A: I do not have access to the paper, but in this literature multiplying a curve by say $2$ roughly means taking the union of the curve with a very close translate of that curve. In reality, the translate actually coincides with the original curve, so we have "twice the same curve". This can be given a precise meaning in the formalism of currents.
