PomeLemma 17.10.28.8A0F90757653 Let $\varphi _0,\varphi _1\in \mathrm{PSH}(X,\theta )$ with $\varphi _1\leq \varphi _0$. Let $(\varphi _ t)_{t\in (0,1)}$ be the geodesic from $\varphi _0$ to $\varphi _1$. Assume that $(\varphi _ t)_{t\in [0,1]}$ is also a subgeodesic. Then $\varphi _ t$ is decreasing in $t\in [0,1]$.
Lemma 17.10.288A0F90757653
Proof. Fix $0< r<s\leq 1$ and put $\delta =(s-r)/(1-r)$. Then $(\varphi _{\delta +(1-\delta )t})_{t\in [0,1]}$ is a subgeodesic from $\varphi _\delta $ to $\varphi _1$. Since
\[ \varphi _\delta \leq (1-\delta )\varphi _0+\delta \varphi _1\leq \varphi _0, \]
we have $\varphi _{\delta +(1-\delta )t}\leq \varphi _ t$ for all $t\in (0,1)$ by the definition of geodesic. In particular, $\varphi _ s=\varphi _{\delta +(1-\delta )r}\leq \varphi _ r$.
It remains to prove that for any $t\in (0,1)$, we have $\varphi _ t\leq \varphi _0$, but this is an immediate consequence of proposition 17.10.7:
\[ \varphi _ t\leq t\varphi _1+(1-t)\varphi _0\leq \varphi _0. \]
$\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.