Lemma 17.10.16.3BEBB86C7F79 Let $(\varphi _ t)_{t\in [0,1]}$ be a subgeodesic in $\mathrm{PSH}(X,\theta )$. Let $0\leq a<b\leq 1$ and let $(\psi _ t)_{t\in [a,b]}$ be a subgeodesic such that $\psi _ a=\varphi _ a$, $\psi _ b=\varphi _ b$, and
\[ \psi _ t\geq \varphi _ t,\quad t\in (a,b). \]
Define
\[ \gamma _ t= \begin{cases} \psi _ t,& \textup{if }t\in (a,b),\\ \varphi _ t,& \textup{if }t\in [0,1]\setminus (a,b). \end{cases} \]
Then $(\gamma _ t)_{t\in [0,1]}$ is a subgeodesic.
Proof.
The $L^1$-convergence of $\gamma _ t$ to $\varphi _0$ at $0$ and $\varphi _1$ at $1$ follows immediately from the corresponding property for $(\varphi _ t)_ t$ and $(\psi _ t)_ t$. It remains to prove that the complexification of $(\gamma _ t)_ t$ is $p_1^*\theta $-psh.
Let
\[ S=\{ z\in \mathbb {C}:0<\operatorname{Re}z<1\} ,\quad S'=\{ z\in \mathbb {C}:a<\operatorname{Re}z<b\} . \]
Let $\Phi $ and $\Psi $ be the complexifications of $(\varphi _ t)_ t$ and $(\psi _ t)_ t$ on the corresponding strips. Since $\Psi \geq \Phi $ on $X\times S'$, the complexification of $(\gamma _ t)_ t$ is equal to $\Phi \lor \Psi $ on $X\times S'$ and to $\Phi $ on $X\times (S\setminus S')$. By the gluing lemma corollary 5.14.20, it is enough to verify that
\[ \varlimsup _{\substack {S'\ni z'\to z\\ x'\to x}}\Psi (x',z')\leq \Phi (x,z) \]
at every point $(x,z)\in X\times S$ with $\operatorname{Re}z=a$ or $\operatorname{Re}z=b$.
Suppose for instance that $a>0$, and let $(x_ j,z_ j)$ tend to $(x,z)$ with $\operatorname{Re}z=a$ and $a<\operatorname{Re}z_ j<b$. Set $t_ j=\operatorname{Re}z_ j$. Since $(\psi _ t)_ t$ is a subgeodesic from $\varphi _ a$ to $\varphi _ b$, the sequence $\psi _{t_ j}$ converges to $\varphi _ a$ in $L^1$. Applying corollary 5.14.8 to this sequence gives
\[ \varphi _ a=\inf _{m>0}\left(\operatorname *{sup}_{j\geq m}{}\! {}^{*}\psi _{t_ j}\right). \]
For fixed $m$, the upper semicontinuity of $\operatorname *{sup}_{j\geq m}{}\! {}^{*}\psi _{t_ j}$ gives
\[ \varlimsup _{j\to \infty }\psi _{t_ j}(x_ j)\leq \left(\operatorname *{sup}_{j\geq m}{}\! {}^{*}\psi _{t_ j}\right)(x). \]
Letting $m\to \infty $, we get
\[ \varlimsup _{j\to \infty }\psi _{t_ j}(x_ j)\leq \varphi _ a(x). \]
Since $\Phi (x,z)=\varphi _ a(x)$ when $\operatorname{Re}z=a$, this is the desired boundary inequality along $\{ \operatorname{Re}z=a\} $. The same argument applies along $\{ \operatorname{Re}z=b\} $ when $b<1$. Hence corollary 5.14.20 shows that the pasted complexification is $p_1^*\theta $-psh.
$\square$
Pome
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