论文标题
离散表面，长度，面积和最小填充圆的表面
Discrete surfaces with length and area and minimal fillings of the circle
论文作者
论文摘要
我们建议想象，表面上的每个riemannian度量在小规模上都是离散的，该曲线由称为壁的曲线制成。曲线的长度是它的壁横梁数量，表面的面积是墙壁本身的交叉数。我们展示了如何通过Wallsystem近似Riemannian（或自我逆向Finsler）指标。 这项工作是由格罗莫夫（Gromov）的填充面积猜想（FAC）激发的，即半球最大程度地减少了可定向的riemannian表面的区域，这些表面填充了一个圆圈。我们引入了一个离散的FAC：每个填充等于$ 2N $循环图的平方孔表面至少具有$ n（n-1）/2 $ squares。我们证明，我们的离散FAC等同于具有自我逆向度量的表面的FAC。 如果表面是磁盘，则离散的fAT遵循Steinitz的算法，将曲线转化为伪线。这为具有自我逆转度量的磁盘的FAC提供了新的证明。我们还使用离散的差异形式模仿了伊万诺夫的同样事实证明。我们证明，该FAC适用于具有自我逆向度量的Möbius频段。为此，我们使用De Graaf-Schrijver和Hass-Scott开发的组合曲线缩短流。通过相同的方法，我们证明了用sabourau-yassine猜想的klein瓶的收缩不平等。 可以使用墙壁离散自我逆转指标，因为每个规范的平面都满足Crofton的公式：每个段的长度等于其横断线的一组线的符合度度量。定向的二维指标没有Crofton公式，但也可以离散。它们的离散化是一个三角剖分，其中每个边的长度以一种方式为1，另一个方式为0，表面的面积是三角形的数量。该结构是一个简单的集合，对偶性图。墙壁的角色是由Postnikov的链扮演的。
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 Möbius 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.
