Discrete surfaces with length and area and minimal fillings of the circle

Abstract

We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian surfaces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrically a $2n$-cycle graph has at least $n(n-1)/2$ squares. We prove that our discrete FAC is equivalent to the FAC for surfaces with self-reverse metric. If the surface is a disk, the discrete FAC follows from Steinitz's algorithm for transforming curves into pseudolines. This gives a new proof of the FAC for disks with self-reverse metric. We also imitate Ivanov's proof of the same fact, using discrete differential forms. And we prove that the FAC holds for M\"obius bands with self-reverse metric. For this we use a combinatorial curve shortening flow developed by de Graaf--Schrijver and Hass--Scott. With the same method we prove the systolic inequality for Klein bottles with self-reverse metric, conjectured by Sabourau--Yassine. Self-reverse metrics can be discretized using walls because every normed plane satisfies Crofton's formula: the length of every segment equals the symplectic measure of the set of lines that it crosses. Directed 2-dimensional metrics have no Crofton formula, but can be discretized as well. Their discretization is a triangulation where the length of each edge is 1 in one way and 0 in the other, and the area of the surface is the number of triangles. This structure is a simplicial set, dual to a plabic graph. The role of the walls is played by Postnikov's strands.