Proof.
Let us begin with a subgeodesic $(\psi _ t)_{t\in (0,1)}$ as in 1. Let $F$ be the convex function as in proposition 21.13.62. We extend $F$ to a function $F\colon N_{\mathbb {R}}\times \mathbb {R}\rightarrow \mathbb {R}$ as follows: For any $n\in N_{\mathbb {R}}$, we define
\[ F(n,t)=\begin{cases} F_{\psi _ t}(n),& \textup{if }0<t<1,\\ F_{\psi _0}(n),& \textup{if }t=0,\\ F_{\psi _1}(n),& \textup{if }t=1,\\ \infty ,& \textup{if } t>1 \textup{ or }t<0. \end{cases} \]
Then $F$ is a proper closed convex function on $N_{\mathbb {R}}\times \mathbb {R}$. Let $\Psi $ be the Legendre transform of $F$. Then $\Psi $ is a proper closed convex function on $M_{\mathbb {R}}\times \mathbb {R}$ by theorem 21.11.24. By (21.11.23), for any $m\in M_{\mathbb {R}}$ and $s\in \mathbb {R}$, we have
\[ \begin{aligned} \Psi (m,s)=& \sup _{n\in N_{\mathbb {R}},t\in [0,1]} \left( \langle m,n\rangle + ts-F(n,t)\right)\\ =& \sup _{t\in [0,1]} \left( ts+F_ t^*(m) \right). \end{aligned} \]
Therefore, the latter half of (21.13.67) follows. Next recall that
\[ \eta \coloneqq \inf _{t\in (0,1)}\psi _ t\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta ), \]
as follows from proposition 17.10.7. Therefore,
\[ \begin{aligned} \Psi (m,s)=& \sup _{n\in N_{\mathbb {R}},t\in [0,1]} \left( \langle m,n\rangle + ts-F(n,t)\right)\\ \leq & \sup _{n\in N_{\mathbb {R}},t\in [0,1]} \left( \langle m,n\rangle + ts-F_{\eta }(n)\right)\\ =& \sup _{t\in [0,1]}ts+G_{\eta }(m)\\ =& (s\lor 0)+G_{\eta }(m). \end{aligned} \]
Conversely, let us begin with a function $\Psi $ as in 2. Let $F$ be the Legendre transform of $\Psi $. We first observe that $F(n,t)=\infty $ for all $n\in N_{\mathbb {R}}$ and $t\not\in [0,1]$. Indeed, the upper bound $\Psi \leq G+(s\lor 0)$ gives
\[ F(n,t)\geq G^*(n)+\sup _{s\in \mathbb {R}}\left(ts-(s\lor 0)\right), \]
and the last supremum is infinite exactly when $t\notin [0,1]$.
For $t\in [0,1]$, the lower bound $\Psi \geq G_{\varphi _0}\lor (G_{\varphi _1}+s)$ implies
\[ \Psi (m,s)\geq (1-t)G_{\varphi _0}(m)+tG_{\varphi _1}(m)+ts. \]
Consequently,
\[ F(n,t)\leq \left((1-t)G_{\varphi _0}+tG_{\varphi _1}\right)^*(n) \leq (1-t)F_{\varphi _0}(n)+tF_{\varphi _1}(n) \]
for all $t\in [0,1]$ and $n\in N_{\mathbb {R}}$. On the other hand, the upper bound $\Psi \leq G+(s\lor 0)$ gives
\[ \begin{aligned} F(n,t)=& \sup _{m\in M_{\mathbb {R}},s\in \mathbb {R}} \left(\langle m,n\rangle +ts-\Psi (m,s)\right)\\ \geq & \sup _{m\in M_{\mathbb {R}},s\in \mathbb {R}} \left(\langle m,n\rangle +ts-G(m)-(s\lor 0)\right)\\ =& G^*(n)+\sup _{s\in \mathbb {R}} \left( ts-(s\lor 0)\right)\\ =& \begin{cases} G^*(n),& \text{if }t\in [0,1],\\ \infty ,& \text{otherwise}. \end{cases}\end{aligned} \]
The convexity of $F$ gives one-sided boundary limits
\[ F_0(n)\coloneqq \lim _{t\to 0+}F(n,t),\quad F_1(n)\coloneqq \lim _{t\to 1-}F(n,t). \]
The estimates above give $F_0\leq F_{\varphi _0}$ and $F_1\leq F_{\varphi _1}$, and also show that $F_0,F_1\in \mathcal{P}(N_{\mathbb {R}},P_ D)$. Let $\psi _0,\psi _1\in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ be the corresponding potentials given by theorem 21.13.53. Then $\psi _0\leq \varphi _0$ and $\psi _1\leq \varphi _1$. Applying proposition 21.13.62 to the restriction of $F$ to $N_{\mathbb {R}}\times (0,1)$, we get a subgeodesic $(\psi _ t)_{t\in (0,1)}$ from $\psi _0$ to $\psi _1$. Thus $(\psi _ t)_{t\in (0,1)}$ satisfies 1. The two operations are inverse to each other by construction.
$\square$
Pome
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