Riemannian and Lorentzian flow-cut theorems

TL;DR

This work establishes geometric max flow–min cut dualities in both Riemannian and Lorentzian settings using convex optimization, Lagrangian duality, and convex relaxation. In the Riemannian case, it proves that the maximum flux of a divergenceless vector field with unit-norm bound through a boundary region equals the minimum area of a homologous bulk surface, and it extends the result to relative homology and nesting. In the Lorentzian setting, it proves a min flow–max cut analogue where a divergenceless timelike flow with norm bound and boundary conditions is dual to the maximal-volume achronal slice homologous to the boundary region, with extensions to Dilworth-type continuum theorems and degenerate metrics. Throughout, the authors emphasize that convex-analytic methods provide a powerful, broadly applicable framework for translating geometric minimization problems into tractable dual problems, with potential applications to holographic entanglement entropy and holographic complexity. The paper thus advances a unified variational perspective on flows, cuts, and their dual geometric objects in both Riemannian and Lorentzian geometries, independent of dynamical equations like Einstein’s equations.

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.

Figures (8)

• Figure 1: Illustration of the homology condition. The bulk surface $m$ is homologous to the boundary region $A$ ($m\sim A$) since there exists a region $r$ (shaded blue) with $\partial r=A-m$. The normal covectors $n_\mu$ to $m$ and $A$ are indicated by the corresponding arrows. (Note that the normal to $m$ is flipped compared to that of $\partial r$, which points outward from the region $r$.)
• Figure 2: Illustration of the level sets $m(p)$ defined below \ref{['rdef']}. The boundary region $A$ is shown in dark blue. The boundary condition for $\psi$ sets it equal to $1$ on $A$ and $0$ on $A^c$. A selection of level sets is shown. The small arrows indicate their orientations. The bulk region $r(p)$ for $p=3/4$ is shown in light blue. As explained below \ref{['rBC']}, the boundary condition on $\psi$ implies that $m(p)\sim A$ for $0<p<1$. As the figure illustrates, for $p>1$, $m(p)\sim\emptyset$, while for $p<0$, $m(p)\sim\partial M$.
• Figure 3: Illustration of relative homology, defined in \ref{['relhomdef']}. The boundary regions $A$ and $R$ are shown in blue and red, respectively. The bulk region $r$ is shown in yellow. The bulk surface $m$ obeys $m\sim A\text{ rel }R$ because \ref{['relhomdef']} is satisfied. The left-hand side of that relation is shown as a dashed curve. (Although it is not shown in this figure, $R$ may also overlap $A$.)
• Figure 4: A simple example where the minimal surface homologous to $A$ does not lie in the interior of $M$. Here $M$ is a region of the flat plane. The true minimal surface homologous to $A$ in this case is simply $A$ itself. Therefore, the infimum of the area over bulk surfaces is not achieved.
• Figure 5: Example of nested minimal surfaces $m(A_s)$ and common max flow $v^*$ for a continuous one-parameter family of nested boundary regions $A_s$. The black curves are the minimal surfaces, and the red curves are the flow lines, or integral curves, of $v^*$. $A_0$ is the point at the top, and $A_1$ is all of $\partial M$. The boundaries $\partial A_s$ foliate $\partial M$, and the surfaces $m(A_s)$ foliate $M$.