An introduction to the $\mathcal F$-completion, the Interface construction and the Gamma set

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I'll start off with a definition that might initially sound pretty abstract and maybe confusing. The point in defining things as I do is to make our lives easier later on. The goal is to focus on a symmetrical subspace of what I'll call the $\mathcal F$-completion. This subspace is called $\mathcal I$. So let's begin:

Let $X$ be a smooth, open, geodesically convex $n$-dimensional manifold with regular polytope boundary, and let $V = \{v_i\}$ be a finite set of vertices in $\partial X$. A block is defined as $\mathcal{B} := X \cup \partial X$, where $\partial X$ denotes the boundary.

A foliational completion (or $\mathcal{F}$-completion) of a block $\mathcal{B}$ is a structure obtained by extending foliations to accumulate at specific vertices in the boundary. Formally, for each pair of distinct vertices $(v_i, v_j) \in V \times V$, there exists a foliation $\mathcal{F}_{v_{ij}}$, whose leaves accumulate at $v_i$ and $v_j$. The $\mathcal{F}$-completion is the union of these foliations:

$$CX_V := \bigcup_{(v_i, v_j) \in V \times V} \mathcal{F}_{v_{ij}}.$$

Each foliation satisfies the condition:

$$\lim_{p \to \partial X} L_\alpha(p) = \{v_i, v_j\},$$

where $L_\alpha$ is a leaf in $\mathcal{F}_{v_{ij}}$.

The last condition ensures that the leaves each accumulate to pairs of vertices.


The Interface construction, $\mathcal I$, (shown above) is a symmetrical subspace of an $\mathcal F$-completion in dimension $n=3$.

The interface construction is just the union of four symmetrical leaves taken from the four foliations which make up the $\mathcal F$-completion. Loosely speaking, the Interface construction has four surfaces, each of which are maximal surfaces of revolution within the unit cube. Here maximal means maximal enclosed volume and we also assume the surfaces have constant Gaussian curvature. 

Now, if we intersect $\mathcal I$ with planes of the form $x_1,x_2,x_3=1/2$ we obtain the following copies of intersection locus of curves:


These are two overlapping "oval" curves with the dark blue region depicting the mutual intersection area. These two curves form a piece of the $\Gamma$-set because they are the intersection locus of curves with the planes $x_3=1/2$.




Here is another diagram showing a piece of the $\Gamma$-set (in red), with $\mathcal I$, and the slicing plane $x_3=1/2$ included (in light green):











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