Classification of good foliations of a pair of pants

Question

The following is a proposition from FLP (Thurston's work on surfaces).

Proposition 6.7 (Classification of good foliations of a pair of pants) The function $\mathcal{MF}_0(P^2)\to\Bbb R^3_+$, which to a good measured foliation $(\mathcal{F},\mu)$ associates the triple $$(m_1,m_2,m_3) = (\mu(\gamma_1),\mu(\gamma_2),\mu(\gamma_3)),$$ induces a bijections of $\mathcal{MF}_0(P^2)$ onto $\Bbb R^3_+-\{0\}$. Here, we think a pair of pants $P^2$ as a disk with two holes. $\gamma_1$ is the outer boundary of a disk and $\gamma_2,\gamma_3$ are the inner boundary of a disk (the index is from left to right).

There're several confusions or figures I can't understand during the proof.
Proof. We begin by describing a right inverse. $\color{red}{\text{The construction depends on the position of the triple with respect to the triangle inequality;}}$ $\color{\red}{\text{to each simplex, we associate one topological configuration.}}$ These are given below for the $6$ types of simplicies.
We remark that if we decompose these figures along the separatrices, we obtain foliated rectangles where the widths (that is, the largest measures of transversals) are determined by the triple. For example in configuration $(1)$, the widths of the $3$ rectangles are: $$\color{red}{\begin{align*} a_{12} & = {1\over 2}(m_1+m_2-m_3)\\ a_{13} & = {1\over 2}(m_1+m_3-m_2)\\ a_{23} & = {1\over 2}(m_2+m_3-m_1). \end{align*}}$$ ... etc.

I basically can't understand those red parts. I mean for the first one, it says triangle inequality but what triangle inequality? $m_1\leq m_2+m_3$? or $m_1+m_2\geq m_3$? What is figure 6.5 in the image talking about? I also can't see where those equalities of the second red part come from.

N.B. I intentionally didn't write up the details of notations or definitions but if someone requires in the comment I'll add.

Answer 1

Figure 6.5 is a picture of the standard $2$-simplex $\Delta \subset \mathbb R^3$, where we regard the coordinates of $\mathbb R^3$ as being identical to the three quantities $m_1,m_2,m_3$: $$\Delta = \{(m_1,m_2,m_3) \, \mid \, m_1,m_2,m_3 \ge 0, m_1+m_2+m_3=1\} $$ You should also be thinking of $\Delta$ as the projectivization of the positive octant $$\mathbb R^3_+ = \{(m_1,m_2,m_2 \, \mid \, m_1,m_2,m_3 \ge 0\} $$ in the sense that the function which maps each ray in $\mathbb R^3_+$ based at $(0,0,0)$ to the unique point of $\Delta$ intersected by that ray is a bijection from the set of such rays to the set $\Delta$.

It helps to add some labels to Figure 6.5:

• The top vertex (to which (6) is pointing) is the intersection of $\Delta$ with the $m_1$ axis, i.e. the ray $m_2=m_3=0$. This is deduced from examining figure 6 which depicts the measured foliation $(\mathcal F,\mu)$ for which $\mu(\gamma_2)=\mu(\gamma_3)=0$.
• The bottom right vertex is the intersection of $\Delta$ with the $m_2$-axis, i.e. the ray $m_1=m_3=0$ and therefore the point to which (4) is pointing is the intersection of $\Delta$ with the ray $m_2=0$, $m_1=m_3$. This is deduced from examining figure (4) which depicts $(\mathcal F,\mu)$ for which $\mu(\gamma_2)=0$ and $\mu(\gamma_1)=\mu(\gamma_3)$.
• By process of elimination, the lower left vertex is the intersection of $\Delta$ with the $m_3$-axis, i.e. the ray $m_1=m_2$.

Perhaps now you can answer some of your own questions regarding $\Delta$ as depicted in Figure 6.5. Here's a few preliminary questions to ask yourself:

• Which portion of $\Delta$ corresponds to the inequality $m_2 + m_3 \le m_1$? (this inequality represents one way that the triangle inequality can fail)
• Which portion of $\Delta$ corresponds to $m_2 + m_3 \ge m_1$? (this inequality represents one of three independent requirements for the triangle inequality to succeed)
• Repeat the above two questions with the indexes $m_1,m_2,m_3$ permuted.
• Which portion of $\Delta$ corresponds to the conjunction of the following three inequalities? \begin{align*} m_3 &\le m_1 + m_2 \\ m_1 &\le m_2 + m_3 \\ m_2 &\le m_3 + m_1 \end{align*} The conjunction of these three inequalities is what we call the triangle inequality on the three quantities $m_1,m_2,m_3$.