Definition 17.7.74A0AFDAD6156

Definition 17.7.7.4A0AFDAD6156 Let $u_0,u_1\in \mathrm{PSH}(X,\theta )$. The subgeodesics whose endpoint potentials are at most $u_0$ and $u_1$, respectively, are called competitors. The Perron segment between $u_0$ and $u_1$ is the regularized upper envelope of the complexifications of all competitors; it is $-\infty $ if there are no competitors. When its endpoint potentials are $u_0$ and $u_1$, it is called the weak geodesic between $u_0$ and $u_1$. We write its slices as $G_ t(u_0,u_1)$, $0<t<1$, and, for a weak geodesic, put $G_0(u_0,u_1)=u_0$ and $G_1(u_0,u_1)=u_1$.