Proof.
We begin with a subgeodesic $(\varphi _ t)_{t\in (0,1)}$ from $\varphi _0$ to $\varphi _1$. Then we define $F\colon N_{\mathbb {R}}\times (0,1)\rightarrow \mathbb {R}$ as follows:
\[ F(n,t)=F_{\varphi _ t}(n). \]
Define $F_ t$ as in (21.13.63), we have
\[ \operatorname{Trop}^* F_ t-\operatorname{Trop}^* F_{\theta }=\varphi _ t,\quad t\in (0,1). \]
Let $t_ j\to 0+$. Since $\varphi _{t_ j}\to \varphi _0$ in $L^1$, after passing to a subsequence we have almost-everywhere convergence on $X$. Restricted to the open torus and using the tropical coordinates, this gives $F_{t_ j}\to F_{\varphi _0}$ almost everywhere on $N_{\mathbb R}$. Since the $F_{t_ j}$'s are convex and locally uniformly bounded above, theorem 21.11.20 upgrades this to pointwise convergence everywhere. As the limit is independent of the subsequence, $F_ t\to F_{\varphi _0}$ everywhere as $t\to 0+$. Similarly, $F_ t\to F_{\varphi _1}$ everywhere as $t\to 1-$.
Next we show that $F$ is convex. Let $p_1\colon X\times S\rightarrow X$ be the projection, where
\[ S\coloneqq \{ z\in \mathbb {C}: \mathrm{e}^{-1}<|z|^2<1\} . \]
Since $(\varphi _ t)_{t\in (0,1)}$ is a subgeodesic, its complexification $\Phi $ is $p_1^*\theta $-psh. Recall that $\Phi $ is defined as
71DDC2004810
21.13.64
\begin{equation} \Phi (x,z)=\operatorname{Trop}^*\left( F_{-\log |z|^2}-F_{\theta } \right)(x). \end{equation}
In particular, $\Psi \colon T(\mathbb {C})\times S\rightarrow \mathbb {R}$ defined by
\[ \Psi (x,z)\coloneqq \Phi (x,z)+\operatorname{Trop}^*F_{\theta }(x)=\operatorname{Trop}^*F_{-\log |z|^2}(x) \]
is plurisubharmonic and $T_ c\times S^1$-invariant. Fix a small enough $\epsilon >0$. We can find a decreasing sequence of $T_ c\times S^1$-invariant plurisubharmonic functions $\Psi _ i$ on $T(\mathbb {C})\times S_{\epsilon }$ converging to $\Psi $ everywhere, where
\[ S_{\epsilon }\coloneqq \{ z\in \mathbb {C}: \mathrm{e}^{-1+\epsilon }<|z|^2<\mathrm{e}^{-\epsilon }\} . \]
Let us write
\[ \Psi _ i(x,z)=\operatorname{Trop}^*F_{i,-\log |z|^2}(x) \]
for some $F_ i\colon N_{\mathbb {R}}\times (\epsilon ,1-\epsilon )\rightarrow \mathbb {R}$.
The same computation as in 21.12.7 shows that $F_ i$ is convex. It follows that $F$, as the decreasing limit of $F_ i$, is also convex on $N_{\mathbb {R}}\times (\epsilon ,1-\epsilon )$. Since $\epsilon >0$ is arbitrary, we conclude that $F$ is convex on $N_{\mathbb {R}}\times (0,1)$.
Conversely, suppose that we are given $F$ in 2. We define $\Phi \colon T(\mathbb {C})\times S\rightarrow \mathbb {R}$ using (21.13.64). The arguments in the previous part can be reversed to show that $\Phi $ is $p_1^*\theta |_{T(\mathbb {C})\times S}$-psh.
By our assumption, for each $t\in (0,1)$, we have
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21.13.65
\begin{equation} F_ t\leq tF_{\varphi _1}+(1-t)F_{\varphi _0}\leq F_{V_{\theta }}+C \end{equation}
for some constant $C\in \mathbb {R}$ independent of the choice of $t$. Therefore, $\Phi $ is bounded from above and hence, by theorem 5.14.10, admits a unique $p_1^*\theta $-psh extension to $X\times S$, which we still denote by $\Phi $. We let
\[ \varphi _ t(x)=\Phi (x,\mathrm{e}^{-t/2}) \]
for all $t\in (0,1)$ and $x\in X$. We claim that $(\varphi _ t)$ is a subgeodesic from $\varphi _0$ to $\varphi _1$.
For this purpose, we only need to show that $(\varphi _ t)_{t\in (0,1)}$ has the correct boundary value. But from our assumption in 2, we know that as $t\to 0+$ (resp. $t\to 1-$), $\varphi _ t\to \varphi _0$ (resp. $\varphi _ t\to \varphi _1$) almost everywhere. In particular, $\sup _ X \varphi _ t\geq -C'$ for some large constant $C'>0$ independent of $t\in (0,1)$. Therefore, together with (21.13.65), we deduce from proposition 5.17.3 that $\{ \varphi _ t\} _{t\in (0,1)}$ is a relatively compact family with respect to the $L^1$-topology. We need to show that each cluster point $\psi $ as $t\to 0+$ is equal to $\varphi _0$. But we already know that $\psi =\varphi _0$ almost everywhere. Hence we deduce $\psi =\varphi _0$ from proposition 5.14.6. As $t\to 0+$, we have $\varphi _ t\xrightarrow {L^1}\varphi _0$. Similarly, as $t\to 1-$, we have $\varphi _ t\xrightarrow {L^1}\varphi _1$.
The two constructions are inverse to each other by construction.
$\square$
Pome
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