On a strictly mean-convex manifold, the least area of a cut in a fixed homology class equals the greatest flux of a unit-bounded divergence-free vector field across it.
The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem
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The max flow/min cut theorem for currents and laminations
On a strictly mean-convex manifold, the least area of a cut in a fixed homology class equals the greatest flux of a unit-bounded divergence-free vector field across it.