Pome

Corollary 17.10.11E31D1E7DDF87

Corollary 17.10.11.E31D1E7DDF87 Let $(\varphi _ t)_{t\in [0,1]}$ be a subgeodesic in $\mathrm{PSH}(X,\theta )$. Then

F238807D334F 17.10.12
\begin{equation} \adjustlimits \sup _ X \inf _{s\in (0,1)}\varphi _ s \leq \sup _ X \varphi _ t\leq \sup _ X \varphi _0\lor \sup _ X \varphi _1 \end{equation}

for all $t\in (0,1)$.

Furthermore, $[0,1] \ni t\mapsto \varphi _ t$ is continuous with respect to the $L^1$-topology.

Proof. (17.10.12) follows immediately from proposition 17.10.7. Let us verify the continuity of $t\mapsto \varphi _ t$. The continuity at $t=0$ or $t=1$ follows from the definition of subgeodesic. While the continuity on $(0,1)$ follows from the convexity of $t\mapsto \varphi _ t$, and the fact that $\varphi _ t\in L^1(X)$ for each $t\in (0,1)$. $\square$

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub