Riemannian and Lorentzian flow-cut theorems

TL;DR

This paper establishes geometric flow-cut theorems in Riemannian and Lorentzian geometries using convex optimization, with applications to holography and partial order theory.

Contribution

It proves new max flow-min cut theorems in both Riemannian and Lorentzian settings, extending previous results and including a continuum Dilworth's theorem.

Findings

Max flow-min cut theorem for boundary regions in Riemannian geometry

Properties of max flow and min cut, including nesting

Lorentzian min flow-max cut theorem relating volume and flux

Abstract

We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous min flow-max cut theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field with norm at least 1. This theorem includes as a special case a continuum version of Dilworth's theorem from the theory of partially ordered sets. We include a brief review of the necessary tools from the theory of convex optimization, in particular Lagrangian duality and convex relaxation.