Corollary 21.13.7123F2FB292586

Proof. First observe that given $g$ as in 2, (21.13.72) indeed induces a geodesic ray in $\mathcal{R}_{\mathrm{tor}}(X,\theta ;\phi )$ thanks to corollary 21.13.68. Therefore, we have a map from 2 to 1.

Conversely, given $\ell \in \mathcal{R}_{\mathrm{tor}}(X,\theta ;\phi )$, it follows from corollary 21.13.68 that $G_{\ell _ t}$ is linear in $t\geq 0$ after restriction to $\operatorname{Int}\{ G_{\phi }<\infty \} $. Let

Note that for $m\in \operatorname{Int}\{ G_{\phi }<\infty \} $, we actually have

Hence, $g'$ is a proper convex function on $M_{\mathbb {R}}$. Define $g=\operatorname{cl}g'$. Then $g$ is a proper closed convex function on $M_{\mathbb {R}}$, and $\overline{\operatorname{Dom}g}=\Delta (\theta ,\phi )$. We have therefore a map from 1 to 2.

It remains to argue that this map is the converse of the preceding map from 2 to 1. The non-trivial point is to verify (21.13.72) holds for the $g$ we just constructed. Fix $t>0$. We need to show that

This holds on $\operatorname{Int}\{ G_{\phi }<\infty \} $ by corollary 21.13.68 and the definition of $g$, but then it holds everywhere thanks to proposition 21.11.18. Our assertion follows. $\square$