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The max flow/min cut theorem for currents and laminations

Aidan Backus

TL;DR

This work develops a continuous max flow/min cut theorem on a compact oriented manifold-with-boundary by pairing divergence-free, $\|F\|_{L^ ablaarrow^ fty}\le 1$ flows with mass-minimizing $d-1$-currents under a relative-homology constraint $[S]=\partial\alpha$. The core result establishes the existence of a minimizing current $C^*$ with $\partial C^*=S$ and $[C^*]=\alpha$ whose mass equals the maximum flux over a dual family $\mathcal{F}$ of admissible vector fields, with a dual maximizer $F^*$; in rational cases, $C^*$ can be realized as a measured oriented lamination, and for $d\le 7$ as a minimal lamination. The proof leans on least-gradient functions on the universal cover to produce a primitive current and a calibration, while handling boundary regularity via mean-curvature barrier conditions. Extensions include anisotropic MFMC via elliptic integrands, a discussion of a Freedman–Headrick discretization, and a Teichmüller-theoretic incarnation due to Thurston, linking duality, laminations, and the stable norm to max-flow/min-cut duality.

Abstract

Motivated by applications to holography and Teichmüller theory, we prove a continuous analogue of the max flow/min cut theorem which also takes the topology of the domain into account.

The max flow/min cut theorem for currents and laminations

TL;DR

This work develops a continuous max flow/min cut theorem on a compact oriented manifold-with-boundary by pairing divergence-free, flows with mass-minimizing -currents under a relative-homology constraint . The core result establishes the existence of a minimizing current with and whose mass equals the maximum flux over a dual family of admissible vector fields, with a dual maximizer ; in rational cases, can be realized as a measured oriented lamination, and for as a minimal lamination. The proof leans on least-gradient functions on the universal cover to produce a primitive current and a calibration, while handling boundary regularity via mean-curvature barrier conditions. Extensions include anisotropic MFMC via elliptic integrands, a discussion of a Freedman–Headrick discretization, and a Teichmüller-theoretic incarnation due to Thurston, linking duality, laminations, and the stable norm to max-flow/min-cut duality.

Abstract

Motivated by applications to holography and Teichmüller theory, we prove a continuous analogue of the max flow/min cut theorem which also takes the topology of the domain into account.
Paper Structure (10 sections, 13 theorems, 50 equations, 1 figure)

This paper contains 10 sections, 13 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some motivating examples
  3. Statement of the main theorem
  4. Acknowledgments
  5. Currents on manifolds-with-boundary
  6. Proof of the max flow/min cut theorem
  7. Epilogue
  8. Anisotropic max flow/min cut theorem
  9. The Freedman--Headrick discretization
  10. Thurston's max flow/min cut theorem

Key Result

Theorem 1.1

In every flow network $(V, E, s_0, s_1)$, where the maximum ranges over flows and the minimum ranges over cuts.

Figures (1)

  • Figure 1: Left: We have to specify a relative homology class to determine how the minimal cut will avoid the "black hole region" $\{x \leq 0\}$, shaded in grey. Right: Because the boundary includes a minimizing cycle, there is no minimal cut.

Theorems & Definitions (23)

  • Theorem 1.1: papadimitriou1982combinatorial
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Corollary 1.5
  • Theorem 2.1: Anzellotti's theorem, Anzellotti1983, BackusBest2
  • Lemma 2.3
  • proof
  • Theorem 2.4: $BV$ trace theorem, Giusti77
  • Theorem 2.5: inverse trace theorem, Górny2024
  • ...and 13 more