What is the least-area surface spanning a given boundary?
In the context of orientable surfaces, this problem is naturally dual to a maximum flow problem, and indeed a classical theorem of Federer says any least-area surface can be calibrated.
Solving the discrete minimum-cost circulation problem in an appropriate cell complex, we can get an arbitrarily good approximation to this continuous problem. But the surfaces we obtained are not discrete minimal in a useful sense.
We consider various notions from discrete differential geometry that can be used to get better discrete minimal surfaces.
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