Theorem 17.7.82FECBE87C487

Proof. The rooftop $w=P_\theta (u_0,u_1)$ belongs to $\mathcal E(X,\theta ,\phi )$, by proposition 17.4.7. The constant curve equal to $w$ is a competitor. By lemma 17.7.6, the Perron segment is therefore a subgeodesic, with endpoint potentials $a_0\leq u_0$ and $a_1\leq u_1$, and $w\leq G_ t\leq (1-t)a_0+ta_1\leq (1-t)u_0+tu_1$. Each slice is at most $\phi +C$ for some constant $C$ and at least $w$, so mass monotonicity puts it in $\mathcal E(X,\theta ,\phi )$.

We show that $a_0=u_0$. For $k\geq 1$ the rooftop $q_ k=P_\theta (u_0,u_1+k)$ is not identically $-\infty $, since it dominates $w$, and the affine curve $q_ k-kt$ is a competitor: its value at $t=0$ is at most $u_0$, and its value at $t=1$ is $q_ k-k\leq u_1$. Hence $G_ t\geq q_ k-kt$ for $0<t<1$, and letting $t\to 0$ in $L^1(X)$ gives $a_0\geq q_ k$ almost everywhere, hence everywhere, for every $k$. By the rooftop representation of proposition 16.25.2 and corollary 17.4.8,

Since $a_0$ is upper semicontinuous and dominates every $q_ k$, it dominates $(\sup _ kq_ k)^*=u_0$; hence $a_0=u_0$. Exchanging the roles of the endpoints, with the competitors $P_\theta (u_0+k,u_1)-k(1-t)$, gives $a_1=u_1$.

Let $[a,b]\subset [0,1]$. The restriction of the weak geodesic to $[a,b]$ is a subgeodesic between the slices $G_ a,G_ b\in \mathcal E(X,\theta ,\phi )$. By the part already proved, the weak geodesic between $G_ a$ and $G_ b$ exists and dominates this restriction. By lemma 17.7.5, replacing the restriction by this weak geodesic gives a subgeodesic on $[0,1]$ with endpoints $u_0,u_1$. It is a competitor for the original Perron segment, so its slices are at most $G_ t$; hence the two coincide on $[a,b]$.

Finally, if $u_0,u_1\in \mathcal E^1(X,\theta ,\phi )$, then $w\in \mathcal E^1(X,\theta ,\phi )$ by proposition 17.4.7, and $w\leq G_ t\preceq \phi $ gives $G_ t\in \mathcal E^1(X,\theta ,\phi )$ by corollary 17.3.17. If $u_0,u_1\in \mathcal E^\infty (X,\theta ,\phi )$, then $\phi -C$ is a common minorant for some $C$, so $G_ t\geq w\geq \phi -C$, while the convex upper bound gives $G_ t\leq \phi +C$. $\square$