TITLE: Max flow, min cut on manifolds
QUESTION [12 upvotes]: If a graph has some half edges marked "input" and some half edges marked "output", it is well known that the smallest number of edges which must be cut to disconnect input from output is equal to the number of edge-disjoint paths from input to output.  In the Riemannian context, say on a closed $d$-manifold, I believe the generalization is that the infimum $(d-1)$-volume of an integral $(d-1)$-cycle ( say submanifold) in a fixed homology class is equal to the supremum of closed $(d-1)$-forms integrated on that class, subject to the constraint that the $L^\infty$ norm of the form is at most 1. This is an $L^1$-$L^\infty$ duality.
I do not know a reference for this and would like to. Does this generalize to all $p < d$ cycles and closed $p$-forms?

REPLY [2 votes]: It's good to read a clearly-stated  geometric version of max-flow min-cut.  I remember reading that in the thesis of John Sullivan, A Crystalline Approximation Theorem for Hypersurfaces.  A linear optimization textbook usually only states the case for graphs.
Since I am here, read the thesis, perhaps I should get on a soapbox and discuss it some when I can.