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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.

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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 $2\Gamma$.

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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.

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  • $\begingroup$ If they coincide how can the minimal surface bounded by that curve be different? $\endgroup$ – Upc Apr 28 at 17:26
  • $\begingroup$ Without (eg topological) restriction, the minimal surface bounding the double of a curve actually has zero area (just take an infinitely thin ribbon). Also taking the same surface doesn't work, because the boundary would be the original curve, not its double (the boundary should go around the curve twice) $\endgroup$ – alesia Apr 29 at 19:15

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