PomeDefinition 17.10.20.774DAAE2CF9E Let $\varphi _0, \varphi _1 \in \mathrm{PSH}(X,\theta )$. The geodesic $(\varphi _ t)_{t\in (0,1)}$ from $\varphi _0$ to $\varphi _1$ is the family of potentials $\varphi _ t\in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $ such that
D0FE4F9BEB1B
17.10.21
\begin{equation} \begin{split} \varphi _ t=\operatorname *{sup}_{}{}\! {}^{*}& \left\{ \psi _ t: (\psi _{s})_ s \textup{ is a subgeodesic from }\psi _0\textup{ to }\psi _1,\right.\\ & \psi _0,\psi _1\in \mathrm{PSH}(X,\theta ),\left.\psi _0\leq \varphi _0,\psi _1\leq \varphi _1\right\} . \end{split} \end{equation}
More generally, let $(\varphi _ t)_{t\in [a,b]}$ ($a,b\in \mathbb {R}$, $a\leq b$) be a curve in $\mathrm{PSH}(X,\theta )$. We say $(\varphi _ t)_{t\in [a,b]}$ is a geodesic if the curve $(\varphi _{tb+(1-t)a})_{t\in (0,1)}$ is a geodesic from $\varphi _ a$ to $\varphi _ b$.
We also say $(\varphi _ t)_{t\in (a,b)}$ is a geodesic in $\mathrm{PSH}(X,\theta )$ from $\varphi _ a$ to $\varphi _ b$.
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.