Proof. Since $\{\unicode[STIX]{x1D703}_{2}\}$ is big we can find $\unicode[STIX]{x1D713}\in \operatorname{PSH}(X,\unicode[STIX]{x1D703}_{2})$ that is smooth in $\operatorname{Amp}(\{\unicode[STIX]{x1D703}_{2}\})$ , $\unicode[STIX]{x1D713}$ has analytic singularities such that $\unicode[STIX]{x1D703}_{2}+dd^{c}\unicode[STIX]{x1D713}\geqslant \unicode[STIX]{x1D700}\unicode[STIX]{x1D714}\geqslant \unicode[STIX]{x1D700}\unicode[STIX]{x1D703}_{1}$ for some $\unicode[STIX]{x1D700}>0$ [Reference BoucksomBou04]. Normalize $\unicode[STIX]{x1D713}$ by $\sup _{X}\unicode[STIX]{x1D713}=0$ and denote by $U=\{\unicode[STIX]{x1D713}>-1\}$ which is a non-empty open subset of $X$ , hence $M_{\unicode[STIX]{x1D703}_{1},U}=\sup _{X}V_{\unicode[STIX]{x1D703}_{1},U}<+\infty$ . Now, the function $u=\unicode[STIX]{x1D713}+\unicode[STIX]{x1D700}V_{\unicode[STIX]{x1D703}_{1},E}^{\ast }$ is $\unicode[STIX]{x1D703}_{2}$ -psh and satisfies $u\leqslant 0$ on $E$ (modulo a pluripolar set). Thus by definition we have $u\leqslant V_{\unicode[STIX]{x1D703}_{2},E}^{\ast }$ . It follows that

$$\begin{eqnarray}M_{\unicode[STIX]{x1D703}_{2},E}\geqslant \sup _{U}u\geqslant \unicode[STIX]{x1D700}\sup _{U}V_{\unicode[STIX]{x1D703}_{1},E}^{\ast }-1.\end{eqnarray}$$

On the other hand $V_{\unicode[STIX]{x1D703}_{1},E}^{\ast }-\sup _{U}V_{\unicode[STIX]{x1D703}_{1},E}^{\ast }$ is $\unicode[STIX]{x1D703}_{1}$ -psh and takes non-positive values on $U$ , hence $V_{\unicode[STIX]{x1D703}_{1},E}^{\ast }-\sup _{U}V_{\unicode[STIX]{x1D703}_{1},E}^{\ast }\leqslant V_{\unicode[STIX]{x1D703}_{1},U}$ . This together with the above inequality yields

$$\begin{eqnarray}M_{\unicode[STIX]{x1D703}_{2},E}\geqslant \unicode[STIX]{x1D700}(M_{\unicode[STIX]{x1D703}_{1},E}-M_{\unicode[STIX]{x1D703}_{1},U})-1,\end{eqnarray}$$

giving that, for some $C>0$ fixed we have $T_{\unicode[STIX]{x1D703}_{2}}(E)\leqslant CT_{\unicode[STIX]{x1D703}_{1}}(E)^{\unicode[STIX]{x1D700}}$ . An elementary calculation using the double estimate of Proposition 2.7 finishes the proof.◻

Proof. Let $\unicode[STIX]{x1D714}$ be a Kähler form on $X$ . By the definition of quasi-continuity, for any $\unicode[STIX]{x1D700}>0$ one can find an open set $U\subset X$ such that all $f_{j}$ are continuous in $X\setminus U$ and $\text{Cap}_{\unicode[STIX]{x1D714}}(U)\leqslant \unicode[STIX]{x1D700}$ . Using Theorem 2.8, a standard argument now gives that

(5) $$\begin{eqnarray}\int _{U}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{n}}\leqslant C\unicode[STIX]{x1D700}^{1/n},\quad \int _{U}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{n}}\leqslant C\unicode[STIX]{x1D700}^{1/n}.\end{eqnarray}$$

Now we can use Tietze’s theorem to extend each $f_{j}|_{X\setminus U},f|_{X\setminus U}$ to a continuous functions $\tilde{f}_{j},\tilde{f}$ on $X$ whose $L^{\infty }$ norm is controlled. It follows from [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Theorem 2.17] that for $j_{0}$ fixed we have

$$\begin{eqnarray}\int _{X}\tilde{f}_{j_{0}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{n}}\rightarrow \int _{X}\tilde{f}_{j_{0}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{n}},\quad \int _{X}\tilde{f}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{n}}\rightarrow \int _{X}\tilde{f}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{n}}.\end{eqnarray}$$

Using (5) and the uniform boundedness of $f_{j},\tilde{f}_{j},f,\tilde{f}$ , we can subsequently write that

$$\begin{eqnarray}\int _{X}f_{j_{0}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{n}}\rightarrow \int _{X}f_{j_{0}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{n}},\quad \int _{X}f\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{n}}\rightarrow \int _{X}f\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{n}}.\end{eqnarray}$$

Finally, using the above and the monotonicity of $f_{j}$ we can write that

$$\begin{eqnarray}\lim _{j_{0}\rightarrow \infty }\int _{X}f_{j_{0}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{n}}\geqslant \lim _{j\rightarrow \infty }\int _{X}f_{j}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}^{n}}\geqslant \int _{X}f\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{1}}\wedge \cdots \wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}^{n}}.\end{eqnarray}$$

After invoking the dominated convergence theorem, the proof is finished. ◻

Proof. Without loss of generality we can assume that $\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\leqslant 0$ . Let $\unicode[STIX]{x1D711}_{j}:=\max (\unicode[STIX]{x1D711},V_{\unicode[STIX]{x1D703}}-j)$ , $\unicode[STIX]{x1D713}_{j}:=\max (\unicode[STIX]{x1D713},V_{\unicode[STIX]{x1D703}}-j)$ be the canonical approximants. For each $j>0$ , it follows from Lemma 2.11 below that there exists a unique $u_{j}\in \operatorname{PSH}(X,\unicode[STIX]{x1D703})$ with minimal singularities such that

(6) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{u_{j}}^{n}=e^{u_{j}-\unicode[STIX]{x1D711}_{j}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}}^{n}+e^{u_{j}-\unicode[STIX]{x1D713}_{j}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{j}}^{n}.\end{eqnarray}$$

Additionally, it follows from Lemma 2.5 (see also the proof of the uniqueness part in Lemma 2.11) that $u_{j}\leqslant \min (\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j})$ . Consequently $u_{j}\leqslant P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j})$ .

Next we claim that

$$\begin{eqnarray}\inf _{j}\int _{X}\unicode[STIX]{x1D712}(u_{j}-V_{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D703}_{u_{j}}^{n}>-\infty .\end{eqnarray}$$

To prove the claim, in view of (6) it suffices to prove that

(7) $$\begin{eqnarray}\inf _{j}\int _{X}\unicode[STIX]{x1D712}(u_{j}-V_{\unicode[STIX]{x1D703}})e^{u_{j}-\unicode[STIX]{x1D711}_{j}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}}^{n}>-\infty .\end{eqnarray}$$

By convexity of $\unicode[STIX]{x1D712}$ it follows that $\unicode[STIX]{x1D712}(t+s)\geqslant \unicode[STIX]{x1D712}(t)+\unicode[STIX]{x1D712}(s)$ , and also $\unicode[STIX]{x1D712}(t)e^{t}\geqslant -C,$ for all $t,s\leqslant 0$ for some $C>0$ . Thus to prove (7) it suffices to check that

$$\begin{eqnarray}\inf _{j}\int _{X}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D711}_{j}-V_{\unicode[STIX]{x1D703}})e^{u_{j}-\unicode[STIX]{x1D711}_{j}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}}^{n}>-\infty .\end{eqnarray}$$

But this holds since $u_{j}\leqslant \unicode[STIX]{x1D711}_{j}$ and $\unicode[STIX]{x1D711}\in {\mathcal{E}}_{\unicode[STIX]{x1D712}}(X,\unicode[STIX]{x1D703})$ . Thus the claim is proved.

Since $\unicode[STIX]{x1D712}(-\infty )=-\infty$ , the claim implies that $\sup _{X}u_{j}$ is uniformly bounded. It thus follows from [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Proposition 2.19] that some subsequence of $u_{j}$ converges in $L^{1}(X,\unicode[STIX]{x1D714}^{n})$ to some $u\in {\mathcal{E}}_{\unicode[STIX]{x1D712}}(X,\unicode[STIX]{x1D703})$ . Since $u_{j}\leqslant P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D711}_{j},\unicode[STIX]{x1D713}_{j})$ it follows that $u\leqslant P_{\unicode[STIX]{x1D703}}(\min (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}))$ . Now, it follows from [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Proposition 2.14] that $P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})\in {\mathcal{E}}(X,\unicode[STIX]{x1D703})$ and so is $P_{[\unicode[STIX]{x1D703},\unicode[STIX]{x1D713}]}(\unicode[STIX]{x1D711})$ .◻

Proof. The uniqueness follows from the Lemma 2.5. Indeed, assume that $\unicode[STIX]{x1D713}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703})$ is another solution, i.e.

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}}^{n}=e^{\unicode[STIX]{x1D713}-u}\unicode[STIX]{x1D703}_{u}^{n}+e^{\unicode[STIX]{x1D713}-v}\unicode[STIX]{x1D703}_{v}^{n}.\end{eqnarray}$$

Set $\unicode[STIX]{x1D707}=e^{u}\unicode[STIX]{x1D703}_{v}^{n}+e^{v}\unicode[STIX]{x1D703}_{u}^{n}$ and $\unicode[STIX]{x1D719}:=u+v$ . Then we can write $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}}^{n}=e^{\unicode[STIX]{x1D711}-\unicode[STIX]{x1D719}}\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}}^{n}=e^{\unicode[STIX]{x1D713}-\unicode[STIX]{x1D719}}\unicode[STIX]{x1D707}$ . It thus follows from Lemma 2.5 that $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}$ .

To prove the existence we approximate $u$ by $u_{j}:=\max (u,-j)$ and note that these are bounded functions. Observe that, for each  $j$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{j}:=e^{-u_{j}}\unicode[STIX]{x1D703}_{u}^{n}+e^{-v_{j}}\unicode[STIX]{x1D703}_{v}^{n}\end{eqnarray}$$

is a non-pluripolar positive measure on $X$ . For each $j>0$ it follows from Theorem 2.3 that there exists $\unicode[STIX]{x1D711}_{j}\in \operatorname{PSH}(X,\unicode[STIX]{x1D703})$ with minimal singularities such that

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}}^{n}=e^{\unicode[STIX]{x1D711}_{j}}\unicode[STIX]{x1D707}_{j}.\end{eqnarray}$$

Let $C>0$ be a constant such that $\sup _{X}|u-v|\leqslant 2C$ . This is possible because $u$ and $v$ have minimal singularities. The function $\unicode[STIX]{x1D719}:=(u+v)/2-C-n\log 2$ is $\unicode[STIX]{x1D703}$ -psh with minimal singularities and it satisfies

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D719}}^{n}\geqslant e^{\unicode[STIX]{x1D719}}\unicode[STIX]{x1D707}_{j}.\end{eqnarray}$$

It thus follows from Lemma 2.5 that $\unicode[STIX]{x1D711}_{j}\geqslant \unicode[STIX]{x1D719}$ for all  $j$ . It also follows from Lemma 2.5 that $\unicode[STIX]{x1D711}_{j}$ is decreasing in  $j$ , the pointwise limit $\unicode[STIX]{x1D711}:=\lim _{j\rightarrow +\infty }\unicode[STIX]{x1D711}_{j}$ has minimal singularities. By continuity of the Monge–Ampère operator (see [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10]) it follows that $\unicode[STIX]{x1D711}$ is the solution we are looking for.◻

Proof. Given $u_{0},u_{1}\in {\mathcal{E}}_{\unicode[STIX]{x1D712}}(X,\unicode[STIX]{x1D703})$ it follows that $P_{\unicode[STIX]{x1D703}}(u_{0},u_{1})\leqslant tu_{0}+(1-t)u_{1}$ , $t\in [0,1]$ . By the above result $P_{\unicode[STIX]{x1D703}}(u_{0},u_{1})\in {\mathcal{E}}_{\unicode[STIX]{x1D712}}(X,\unicode[STIX]{x1D703})$ . Now [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Proposition 2.14] implies that $tu_{0}+(1-t)u_{1}\in {\mathcal{E}}_{\unicode[STIX]{x1D712}}(X,\unicode[STIX]{x1D703})$ .◻

Proof. Without loss of generality we can assume that $f\leqslant 0$ . If $f$ is smooth, then the result follows from a balayage argument (or directly from Theorem 2.6). To treat the general case, we approximate $f$ from above (by semicontinuity) by a sequence of smooth functions $(f_{j})$ . We can also assume that $f_{j}\leqslant 0$ . Set $\unicode[STIX]{x1D711}_{j}:=P_{\unicode[STIX]{x1D703}}(f_{j})$ , $\unicode[STIX]{x1D711}:=P_{\unicode[STIX]{x1D703}}(f)$ and note that $\unicode[STIX]{x1D711}_{j}{\searrow}\unicode[STIX]{x1D711}$ . For each $j\in \mathbb{N}$ the measure $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}}^{n}$ vanishes in the set $\{\unicode[STIX]{x1D711}_{j}<f_{j}\}$ . Now, we want to pass to the limit as $j\rightarrow +\infty$ . We first fix $k,l\in \mathbb{N}$ and set

$$\begin{eqnarray}U_{k,l}=\{\unicode[STIX]{x1D711}_{k}<f\}\cap \{\unicode[STIX]{x1D711}>V_{\unicode[STIX]{x1D703}}-l\}.\end{eqnarray}$$

For any $j>k$ , note that on $U_{k,l}$ we have $\unicode[STIX]{x1D711}_{j}\geqslant \unicode[STIX]{x1D711}>V_{\unicode[STIX]{x1D703}}-l$ and $\{\unicode[STIX]{x1D711}_{k}<f\}\subset \{\unicode[STIX]{x1D711}_{j}<f_{j}\}$ . It thus follows from definition of the non-pluripolar product (see [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10]) that for any $j>k$ , the measure $\unicode[STIX]{x1D703}_{\max (\unicode[STIX]{x1D711}_{j},V_{\unicode[STIX]{x1D703}}-l)}^{n}=\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{j}}^{n}$ vanishes on $U_{k,l}$ . By assumption $f$ is quasi-continuous, hence $U_{k,l}$ is quasi open. More precisely, for any fixed $\unicode[STIX]{x1D700}>0$ there exists an open set $V_{\unicode[STIX]{x1D700}}$ such that the set $G_{\unicode[STIX]{x1D700}}:=(V_{\unicode[STIX]{x1D700}}\setminus U_{k,l}\cup U_{k,l}\setminus V_{\unicode[STIX]{x1D700}})$ satisfies $\operatorname{Cap}_{\unicode[STIX]{x1D714}}(G_{\unicode[STIX]{x1D700}})\leqslant \unicode[STIX]{x1D700}$ . Observe that for fixed $l$ all the functions $\max (\unicode[STIX]{x1D711}_{j},V_{\unicode[STIX]{x1D703}}-l)$ are sandwiched between $V_{\unicode[STIX]{x1D703}}-l$ and $V_{\unicode[STIX]{x1D703}}$ . It then follows that

$$\begin{eqnarray}\sup _{j\in \mathbb{N}}\int _{G_{\unicode[STIX]{x1D700}}}\unicode[STIX]{x1D703}_{\max (\unicode[STIX]{x1D711}_{j},V_{\unicode[STIX]{x1D703}}-l)}^{n}\leqslant A\operatorname{Cap}_{\unicode[STIX]{x1D703}}(G_{\unicode[STIX]{x1D700}})\leqslant A^{\prime }\operatorname{Cap}_{\unicode[STIX]{x1D714}}^{1/n}(G_{\unicode[STIX]{x1D700}})\leqslant A^{\prime }\unicode[STIX]{x1D700}^{1/n},\end{eqnarray}$$

where the last inequality follows from the comparison of capacities in Theorem 2.8. Consequently, $\sup _{j\in \mathbb{N}}\int _{V_{\unicode[STIX]{x1D700}}}\unicode[STIX]{x1D703}_{\max (\unicode[STIX]{x1D711}_{j},V_{\unicode[STIX]{x1D703}}-l)}^{n}\leqslant A^{\prime }\unicode[STIX]{x1D700}^{1/n}$ , and the continuity of the Monge–Ampère operator allows to take the limit, and we ultimately obtain

$$\begin{eqnarray}\int _{U_{k,l}}\unicode[STIX]{x1D703}_{\max (\unicode[STIX]{x1D711},V_{\unicode[STIX]{x1D703}}-l)}^{n}\leqslant C\unicode[STIX]{x1D700}^{1/n},\end{eqnarray}$$

for some positive constant $C$ independent of $\unicode[STIX]{x1D700}$ (but dependent on  $l$ ). Now letting $\unicode[STIX]{x1D700}\rightarrow 0$ we see that $\unicode[STIX]{x1D703}_{\max (\unicode[STIX]{x1D711},V_{\unicode[STIX]{x1D703}}-l)}^{n}$ vanishes in $U_{k,l}$ . Letting $l\rightarrow +\infty$ , and using the definition of the non-pluripolar product, we see that $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}}^{n}$ vanishes in $\{\unicode[STIX]{x1D711}_{k}<f\}$ . Now, letting $k\rightarrow +\infty$ we obtain the result.◻

Proof. For each $t>0$ , since $\min (\unicode[STIX]{x1D711}+t,\unicode[STIX]{x1D713})$ is usc and quasi-continuous, it follows from Proposition 2.13 that $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}$ vanishes on $\{\unicode[STIX]{x1D713}_{t}<\min (\unicode[STIX]{x1D711}+t,\unicode[STIX]{x1D713})\}$ , where $\unicode[STIX]{x1D713}_{t}:=P_{\unicode[STIX]{x1D703}}(\min (\unicode[STIX]{x1D711}+t,\unicode[STIX]{x1D713}))$ . Because $\{\unicode[STIX]{x1D713}_{t}<\unicode[STIX]{x1D713}<\unicode[STIX]{x1D711}+t\}\subset \{\unicode[STIX]{x1D713}_{t}<\min (\unicode[STIX]{x1D711}+t,\unicode[STIX]{x1D713})\}$ it thus follows that

$$\begin{eqnarray}\int _{\{\unicode[STIX]{x1D713}_{t}<\unicode[STIX]{x1D713}<\unicode[STIX]{x1D711}+t\}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}=0.\end{eqnarray}$$

Fix $s>0,j>0$ and set $v:=P_{[\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}]}(\unicode[STIX]{x1D713})=(\lim _{t\rightarrow +\infty }\unicode[STIX]{x1D713}_{t})^{\ast }$ , $\unicode[STIX]{x1D713}_{t,j}:=\max (\unicode[STIX]{x1D713}_{t},V_{\unicode[STIX]{x1D703}}-j)$ , $v_{j}:=\,\max (v,V_{\unicode[STIX]{x1D703}}-j)$ . It is clear that $\unicode[STIX]{x1D713}_{t,j}{\nearrow}v_{j}$ almost everywhere as $t{\nearrow}\infty$ . By the plurifine property of the non-pluripolar Monge–Ampère measure it follows that

$$\begin{eqnarray}\int _{\{V_{\unicode[STIX]{x1D703}}-j<\unicode[STIX]{x1D713}_{t}<\unicode[STIX]{x1D713}<\unicode[STIX]{x1D711}+t\}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t,j}}^{n}=0.\end{eqnarray}$$

For $t>s$ , we have $\unicode[STIX]{x1D713}_{s}\leqslant \unicode[STIX]{x1D713}_{t}\leqslant v\leqslant \unicode[STIX]{x1D713}$ . Consequently, $\{V_{\unicode[STIX]{x1D703}}-j<\unicode[STIX]{x1D713}_{s}\leqslant v<\unicode[STIX]{x1D713}<\unicode[STIX]{x1D711}+s\}\subset \{V_{\unicode[STIX]{x1D703}}-j<\unicode[STIX]{x1D713}_{t}<\unicode[STIX]{x1D713}<\unicode[STIX]{x1D711}+t\}$ and we have

$$\begin{eqnarray}\int _{\{V_{\unicode[STIX]{x1D703}}-j<\unicode[STIX]{x1D713}_{s}\leqslant v<\unicode[STIX]{x1D713}<\unicode[STIX]{x1D711}+s\}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t,j}}^{n}=0.\end{eqnarray}$$

Now, using the same trick as in the proof of Proposition 2.13 we let $t\rightarrow +\infty$ and arrive at

$$\begin{eqnarray}\int _{\{V_{\unicode[STIX]{x1D703}}-j<\unicode[STIX]{x1D713}_{s}\leqslant v<\unicode[STIX]{x1D713}<\unicode[STIX]{x1D711}+s\}}\unicode[STIX]{x1D703}_{v_{j}}^{n}=0.\end{eqnarray}$$

Letting $s\rightarrow +\infty$ , then $j\rightarrow +\infty$ , and noting that $\unicode[STIX]{x1D703}_{v}^{n}$ is a non-pluripolar measure, we can conclude that $\unicode[STIX]{x1D703}_{v}^{n}$ vanishes on $\{v<\unicode[STIX]{x1D713}\}$ . This concludes the proof.◻

Proof. It follows from Theorem 2.10 that $P_{[\unicode[STIX]{x1D703},\unicode[STIX]{x1D711}]}(\unicode[STIX]{x1D713})\in {\mathcal{E}}(X,\unicode[STIX]{x1D703})$ . The conclusion then follows from Proposition 2.14 and the domination principle (Proposition 2.4).◻

Proof. The proof is essentially the same as in [Reference BerndtssonBer15, § 2.2], so we only point out the ideas. Consider the following (barrier) subgeodesic of $\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$ :

$$\begin{eqnarray}u_{t}(x):=\max (\unicode[STIX]{x1D711}_{0}-Ct,\unicode[STIX]{x1D711}_{1}+C(t-1)),\end{eqnarray}$$

where $C$ is a positive constant such that $\unicode[STIX]{x1D711}_{1}-C\leqslant \unicode[STIX]{x1D711}_{0}\leqslant \unicode[STIX]{x1D711}_{1}+C$ . It is clear that $[0,1]\ni t\rightarrow u_{t}\in \operatorname{PSH}(X,\unicode[STIX]{x1D703})$ is a subgeodesic of $\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$ , and by $t$ -convexity of $t\rightarrow \unicode[STIX]{x1D711}_{t}$ we can write $u_{t}\leqslant \unicode[STIX]{x1D711}_{t}\leqslant (1-t)\unicode[STIX]{x1D711}_{0}+t\unicode[STIX]{x1D711}_{1}$ , hence the conclusion about Lipschitz continuity of $t\rightarrow \unicode[STIX]{x1D711}_{t}$ follows.

The proof of (10) follows from a standard balayage argument and we refer the interested reader to [Reference BerndtssonBer15, § 2.2], to see how the ideas from there generalize to our setting. ◻

Proof. Fix $\unicode[STIX]{x1D700}>0$ , $\unicode[STIX]{x1D6FF}>0$ . As $\unicode[STIX]{x1D703}$ is big we can find $\unicode[STIX]{x1D713}\in \operatorname{PSH}(X,\unicode[STIX]{x1D703})$ , $\sup _{X}\unicode[STIX]{x1D713}=0$ , with analytic singularities such that $X\setminus \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})=\{\unicode[STIX]{x1D713}=-\infty \}$ and $\unicode[STIX]{x1D713}\leqslant u_{s},v_{s},~s\in D$ , such that $\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D713}$ dominates a Kähler form. Consider

$$\begin{eqnarray}u_{\unicode[STIX]{x1D700}}:=\max (u,v_{\unicode[STIX]{x1D700}}),\quad v_{\unicode[STIX]{x1D700}}:=(1-\unicode[STIX]{x1D6FF})v+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}\unicode[STIX]{x1D713}-\unicode[STIX]{x1D700},\end{eqnarray}$$

for some constant $\unicode[STIX]{x1D706}>1$ . If $\unicode[STIX]{x1D706}-1$ is small enough we have

$$\begin{eqnarray}\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D706}\unicode[STIX]{x1D713}=\unicode[STIX]{x1D706}(\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D713})+(1-\unicode[STIX]{x1D706})\unicode[STIX]{x1D703}\geqslant \unicode[STIX]{x1D706}\unicode[STIX]{x1D714}+(1-\unicode[STIX]{x1D706})\unicode[STIX]{x1D703}\geqslant 0,\end{eqnarray}$$

where $\unicode[STIX]{x1D714}$ is a Kähler form on $X$ . Thus $v_{\unicode[STIX]{x1D700}}$ is $\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}$ -psh on $X\times D$ . Observe that for some open relatively compact $\unicode[STIX]{x1D6FA}^{\prime }\Subset \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})$ ( $\unicode[STIX]{x1D6FA}^{\prime }$ depends on $\unicode[STIX]{x1D706}$ ), $K\Subset D$ , we have $u_{\unicode[STIX]{x1D700}}\equiv u$ in a neighborhood containing $(X\setminus \unicode[STIX]{x1D6FA}^{\prime })\times (D\setminus K)$ . It follows that $\int _{Y}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u)^{n+1}=\int _{Y}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u_{\unicode[STIX]{x1D700}})^{n+1}$ , where $Y:=\unicode[STIX]{x1D6FA}^{\prime }\times K$ . Indeed, for any test function $0\leqslant \unicode[STIX]{x1D712}\in {\mathcal{C}}^{\infty }(Y)$ which is identically $1$ in an open neighborhood $U$ inside $Y$ such that $\{u<u_{\unicode[STIX]{x1D700}}\}\subset U$ we have

$$\begin{eqnarray}\int _{Y}\unicode[STIX]{x1D712}[(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u)^{n+1}-(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u_{\unicode[STIX]{x1D700}})^{n+1}]=\int _{Y}(u-u_{\unicode[STIX]{x1D700}})dd^{c}\unicode[STIX]{x1D712}\wedge T=0,\end{eqnarray}$$

where $T$ is a positive $(n,n)$ -current.

Recall that for locally bounded $\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}$ -psh functions $\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}$ , the maximum principle for the complex Monge–Ampère operator yields

$$\begin{eqnarray}\mathbf{1}_{\{\unicode[STIX]{x1D711}<\unicode[STIX]{x1D713}\}}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}\max (\unicode[STIX]{x1D711},\unicode[STIX]{x1D713}))^{n+1}=\mathbf{1}_{\{\unicode[STIX]{x1D711}<\unicode[STIX]{x1D713}\}}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D713})^{n+1}.\end{eqnarray}$$

Using the above facts we can write

$$\begin{eqnarray}\displaystyle \int _{\{u<v_{\unicode[STIX]{x1D700}}\}\cap \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}v_{\unicode[STIX]{x1D700}})^{n+1} & = & \displaystyle \int _{\{u<v_{\unicode[STIX]{x1D700}}\}\cap \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u_{\unicode[STIX]{x1D700}})^{n+1}\nonumber\\ \displaystyle & = & \displaystyle \int _{Y}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u_{\unicode[STIX]{x1D700}})^{n+1}-\int _{\{u\geqslant v_{\unicode[STIX]{x1D700}}\}\cap Y}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u_{\unicode[STIX]{x1D700}})^{n+1}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \int _{Y}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u)^{n+1}-\int _{\{u>v_{\unicode[STIX]{x1D700}}\}\cap Y}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u_{\unicode[STIX]{x1D700}})^{n+1}\nonumber\\ \displaystyle & = & \displaystyle \int _{Y}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u)^{n+1}-\int _{\{u>v_{\unicode[STIX]{x1D700}}\}\cap Y}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u)^{n+1}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \int _{\{u<(1-\unicode[STIX]{x1D6FF})v+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}\unicode[STIX]{x1D713}-\unicode[STIX]{x1D700}\}\cap \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}u)^{n+1}.\nonumber\end{eqnarray}$$

The left-hand side in the above estimate can be further estimated in the following way:

$$\begin{eqnarray}\int _{\{u<v_{\unicode[STIX]{x1D700}}\}\cap \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}v_{\unicode[STIX]{x1D700}})^{n+1}\geqslant (1-\unicode[STIX]{x1D6FF})^{n}\int _{\{u<v_{\unicode[STIX]{x1D700}}\}\cap \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}v)^{n+1}.\end{eqnarray}$$

Letting $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D700}\rightarrow 0$ , the proof is finished.◻

Proof. Let $t\rightarrow \unicode[STIX]{x1D711}_{t}$ be the weak geodesic connecting $\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$ . It suffices to show that $\unicode[STIX]{x1D6F7}\leqslant U$ . Fix $\unicode[STIX]{x1D70C}$ a smooth function in $D$ such that $\unicode[STIX]{x1D70C}=0$ on the boundary and $dd^{c}\unicode[STIX]{x1D70C}=\sqrt{-1}\,dz\wedge d\bar{z}$ . Fix also $\unicode[STIX]{x1D70F}\in \operatorname{PSH}(X,\unicode[STIX]{x1D703})$ with minimal singularities such that $(\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D70F})^{n}=c\unicode[STIX]{x1D714}^{n}$ , for some positive normalization constant $c$ and some fixed Kähler form $\unicode[STIX]{x1D714}$ on  $X$ . Observe that such a potential exists thanks to [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Theorem 4.1]. We normalize $\unicode[STIX]{x1D70F}$ so that $\unicode[STIX]{x1D70F}\leqslant \min (\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1})$ . Applying the comparison principle (Proposition 3.2) for $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}:=(1-\unicode[STIX]{x1D700})\unicode[STIX]{x1D6F7}+\unicode[STIX]{x1D700}(\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D70F})$ and $U$ we get

$$\begin{eqnarray}\int _{\{U<\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}\}\cap \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}})^{n+1}\leqslant \int _{\{U<\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}\}\cap \operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}U)^{n+1}=0.\end{eqnarray}$$

By expanding $(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}})^{n+1}\geqslant \unicode[STIX]{x1D700}^{n}(\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D70F})^{n}\wedge dd^{c}\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D700}^{n}c\unicode[STIX]{x1D714}^{n}\wedge dd^{c}\unicode[STIX]{x1D70C}>0$ , the inequality above gives that $U\geqslant \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}$ almost everywhere, hence everywhere in $X\times D$ . Letting $\unicode[STIX]{x1D700}\rightarrow 0$ we get the desired result.◻

Proof. The fact that $t\rightarrow \unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700},t}$ is a subgeodesic is trivial. Because $t\rightarrow \unicode[STIX]{x1D711}_{t}$ is $t$ -Lipschitz, it follows that $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700},t}$ has minimal singularities for all $t\in [\unicode[STIX]{x1D700},1-\unicode[STIX]{x1D700}]$ . Working directly with (11) we obtain $|\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700},t}-\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700},t^{\prime }}|\leqslant C|t-t^{\prime }|$ .◻

Proof. Write $\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}^{\prime }(t)=\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{+}(t)-\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{-}(t)$ , where $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{+},\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{-}$ are the positive and negative parts of $\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}^{\prime }(t)$ . They are clearly bounded (but the bound blows up as $\unicode[STIX]{x1D700}\rightarrow 0$ ). Now, let

$$\begin{eqnarray}u_{\unicode[STIX]{x1D700},t}^{\pm }(x):=\int _{0}^{1}\unicode[STIX]{x1D711}_{s}(x)\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{\pm }(t-s)\,ds.\end{eqnarray}$$

It follows that

$$\begin{eqnarray}dd^{c}u_{\unicode[STIX]{x1D700},t}^{\pm }=\int _{0}^{1}(dd^{c}\unicode[STIX]{x1D711}_{s})\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{\pm }(t-s)\,ds\geqslant -A_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D703},\end{eqnarray}$$

where $A_{\unicode[STIX]{x1D700}}=\int _{-\unicode[STIX]{x1D700}}^{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{+}=\int _{-\unicode[STIX]{x1D700}}^{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{-}$ . Observe indeed that $\int _{-\unicode[STIX]{x1D700}}^{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{+}-\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{-}=\int _{-\unicode[STIX]{x1D700}}^{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}^{\prime }=0$ . To see that $u_{\unicode[STIX]{x1D700},t}^{\pm }/A_{\unicode[STIX]{x1D700}}$ has minimal singularities we note that $|\unicode[STIX]{x1D711}_{t}-\unicode[STIX]{x1D711}_{t^{\prime }}|\leqslant C|t-t^{\prime }|$ , hence

$$\begin{eqnarray}u_{\unicode[STIX]{x1D700},t}^{\pm }(x):=\int _{-\unicode[STIX]{x1D700}}^{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{t-s}(x)\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{\pm }(s)\,ds\leqslant \int _{-\unicode[STIX]{x1D700}}^{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D711}_{t^{\prime }-s}(x)\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{\pm }(s)\,ds+CA_{\unicode[STIX]{x1D700}}|t-t^{\prime }|.\end{eqnarray}$$

For $\ddot{\unicode[STIX]{x1D711}}_{\unicode[STIX]{x1D700},t}$ , a similar argument works with the choice $B_{\unicode[STIX]{x1D700}}:=\int _{-\unicode[STIX]{x1D700}}^{\unicode[STIX]{x1D700}}\max (\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}^{\prime \prime }(t),0)\,dt$ .◻

Proof. For notational convenience we remove the subscript $\unicode[STIX]{x1D700}$ . By basic properties of the $\text{I}$ functional (see [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Reference Berman and BoucksomBB10]) we have, for $t\in (\unicode[STIX]{x1D700},1-\unicode[STIX]{x1D700}),s>0$ ,

$$\begin{eqnarray}\int _{X}(\unicode[STIX]{x1D711}_{t+s}-\unicode[STIX]{x1D711}_{t})\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t+s}}^{n}\leqslant \text{I}(\unicode[STIX]{x1D711}_{t+s})-\text{I}(\unicode[STIX]{x1D711}_{t})\leqslant \int _{X}(\unicode[STIX]{x1D711}_{t+s}-\unicode[STIX]{x1D711}_{t})\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}.\end{eqnarray}$$

Dividing by $s>0$ the right-hand side then converges (as $s\rightarrow 0$ ) to $\int _{X}\dot{\unicode[STIX]{x1D711}}_{t}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}$ . The left-hand side can be estimated by

$$\begin{eqnarray}\frac{1}{s}\int _{X}(\unicode[STIX]{x1D711}_{t+s}-\unicode[STIX]{x1D711}_{t})\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t+s}}^{n}\geqslant \int _{X}\dot{\unicode[STIX]{x1D711}}_{t}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t+s}}^{n}.\end{eqnarray}$$

As $\dot{\unicode[STIX]{x1D711}}_{t}$ is bounded and quasi-continuous (with respect to the Monge–Ampère capacity $\operatorname{Cap}_{\unicode[STIX]{x1D714}}$ ), Corollary 2.9 allows to pass to the limit as $s\rightarrow 0$ and the differentiability of $\text{I}(\unicode[STIX]{x1D711}_{t})$ follows.

To prove the formula for the second derivative we fix $t\in (\unicode[STIX]{x1D700},1-\unicode[STIX]{x1D700})$ and $s>0$ small enough and prove that

$$\begin{eqnarray}\lim _{s\rightarrow 0^{+}}d(s):=\lim _{s\rightarrow 0^{+}}\frac{1}{s}\biggl(\int _{X}\dot{\unicode[STIX]{x1D711}}_{t+s}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t+s}}^{n}-\dot{\unicode[STIX]{x1D711}}_{t}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}\biggr)\end{eqnarray}$$

equals to the right-hand side of (12). The same formula for the left limit will then follow automatically, as we can always ‘reverse’ the time direction of a subgeodesic. Setting $T_{s}:=\sum _{j=1}^{n-1}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{j}\wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t+s}}^{n-1-j}$ , we can write

$$\begin{eqnarray}d(s)=\frac{1}{s}\biggl(\int _{X}\dot{\unicode[STIX]{x1D711}}_{t+s}dd^{c}(\unicode[STIX]{x1D711}_{t+s}-\unicode[STIX]{x1D711}_{t})\wedge T_{s}+\int _{X}(\dot{\unicode[STIX]{x1D711}}_{t+s}-\dot{\unicode[STIX]{x1D711}}_{t})\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}\biggr).\end{eqnarray}$$

Thanks to [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Theorem 1.14] we can integrate by parts in the first term and obtain

(13) $$\begin{eqnarray}d(s)=\frac{1}{s}\biggl(\int _{X}(\unicode[STIX]{x1D711}_{t+s}-\unicode[STIX]{x1D711}_{t})dd^{c}\dot{\unicode[STIX]{x1D711}}_{t+s}\wedge T_{s}+\int _{X}(\dot{\unicode[STIX]{x1D711}}_{t+s}-\dot{\unicode[STIX]{x1D711}}_{t})\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}\biggr).\end{eqnarray}$$

As $s\rightarrow 0$ , $(\unicode[STIX]{x1D711}_{t+s}-\unicode[STIX]{x1D711}_{t})/s$ decreases to $\dot{\unicode[STIX]{x1D711}}_{t}$ , all of them being quasi-continuous and uniformly bounded. On the other hand, by Lemma 3.5 we can write $\dot{\unicode[STIX]{x1D711}}_{t+s}/A_{\unicode[STIX]{x1D700}}=u_{t+s}^{+}-u_{t+s}^{-}$ , where $u_{t+s}^{\pm }$ are $\unicode[STIX]{x1D703}$ -psh functions with minimal singularities that converge uniformly to $u_{t}^{\pm }$ as $s\rightarrow 0$ . By using Corollary 2.9, the first term in the right-hand side of (13) converges to $n\int _{X}\dot{\unicode[STIX]{x1D711}}_{t}dd^{c}\dot{\unicode[STIX]{x1D711}}_{t}\wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n-1}$ . Moreover, using dominated convergence for the second term we obtain

$$\begin{eqnarray}\lim _{s\rightarrow 0^{+}}\frac{1}{s}\int _{X}(\dot{\unicode[STIX]{x1D711}}_{t+s}-\dot{\unicode[STIX]{x1D711}}_{t})\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}=\int _{X}\ddot{\unicode[STIX]{x1D711}}_{t}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}.\end{eqnarray}$$

Observe in fact that $\ddot{\unicode[STIX]{x1D711}}_{t}$ is uniformly bounded from above as it can be written as the difference of two quasi-plurisubharmonic functions. Finally, an integration by parts gives

$$\begin{eqnarray}\lim _{s\rightarrow 0^{+}}d(s)=-n\int _{X}\,d\dot{\unicode[STIX]{x1D711}}_{t}\wedge d^{c}\dot{\unicode[STIX]{x1D711}}_{t}\wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n-1}+\int _{X}\ddot{\unicode[STIX]{x1D711}}_{t}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}.\Box\end{eqnarray}$$

Proof. We again drop the $\unicode[STIX]{x1D700}$ subscript and fix $t\in (\unicode[STIX]{x1D700},1-\unicode[STIX]{x1D700})$ for the duration of the proof. In view of Lemma 3.6 it is enough to prove that

(14) $$\begin{eqnarray}-n\,d\dot{\unicode[STIX]{x1D711}}_{t}\wedge d^{c}\dot{\unicode[STIX]{x1D711}}_{t}\wedge \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n-1}+\ddot{\unicode[STIX]{x1D711}}_{t}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D711}_{t}}^{n}\geqslant 0\end{eqnarray}$$

in the weak sense of measures in $\operatorname{Amp}(\{\unicode[STIX]{x1D703}\})$ . This property is local, hence we can work in relatively compact open subset $K$ of $\operatorname{Amp}(\{\unicode[STIX]{x1D703}\})$ , and approximate $\unicode[STIX]{x1D711}_{t}(x)$ using a convolution in the space variable $x$ . As we show now, since $\dot{\unicode[STIX]{x1D711}}_{t}$ , $\ddot{\unicode[STIX]{x1D711}}_{t}$ are bounded in $K$ and can be written as the difference of two quasi-psh functions with minimal singularities, the convergence of the appropriate approximating measures to the left-hand side in (14) follows from standard pluripotential theory. Indeed, fix a coordinate ball $B\subset K$ . We will show that (14) holds in $B$ in the weak sense of measures.

We can assume existence of a smooth local potential $\unicode[STIX]{x1D70F}\in C^{\infty }(B)$ such that $\unicode[STIX]{x1D703}=dd^{c}\unicode[STIX]{x1D70F}$ . Let $\tilde{\unicode[STIX]{x1D70C}}_{\unicode[STIX]{x1D6FF}}(x)$ be a smoothing kernel in $\mathbb{C}^{n}$ and consider

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D711}}_{t}^{\unicode[STIX]{x1D6FF}}(x):=\int _{B}(\unicode[STIX]{x1D70F}(x-\unicode[STIX]{x1D701})+\unicode[STIX]{x1D711}_{t}(x-\unicode[STIX]{x1D701}))\tilde{\unicode[STIX]{x1D70C}}_{\unicode[STIX]{x1D6FF}}(\unicode[STIX]{x1D701})\,dV(\unicode[STIX]{x1D701})-\unicode[STIX]{x1D70F}(x).\end{eqnarray}$$

Since the complexification of $\tilde{\unicode[STIX]{x1D711}}_{t}^{\unicode[STIX]{x1D6FF}}$ is smooth and $\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}$ -psh in $B_{\unicode[STIX]{x1D6FF}}\times D_{\unicode[STIX]{x1D700}}$ , it follows that (14) holds for $\tilde{\unicode[STIX]{x1D711}}_{t}^{\unicode[STIX]{x1D6FF}}$ and $\tilde{\unicode[STIX]{x1D711}}_{t}^{\unicode[STIX]{x1D6FF}}{\searrow}\unicode[STIX]{x1D711}_{t}$ .

By Lemma 3.5 we can write $\dot{\unicode[STIX]{x1D711}}_{t}:=u_{+}-u_{-}$ , where $u_{+},u_{-}$ are bounded quasi psh functions on $B_{\unicode[STIX]{x1D6FF}}$ . Then the corresponding smooth quasi psh functions $v_{\pm }^{\unicode[STIX]{x1D6FF}}$ , defined by

$$\begin{eqnarray}v_{\pm }^{\unicode[STIX]{x1D6FF}}(x):=\int _{0}^{1}\int _{B}(\unicode[STIX]{x1D70F}(x-\unicode[STIX]{x1D701})+\unicode[STIX]{x1D711}_{s}(x-\unicode[STIX]{x1D701}))\tilde{\unicode[STIX]{x1D70C}}_{\unicode[STIX]{x1D6FF}}(\unicode[STIX]{x1D701})\,\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D700}}^{\pm }(t-s)\,ds\,dV(\unicode[STIX]{x1D701}),\end{eqnarray}$$

converge decreasingly to $u_{\pm }$ and we have $\dot{\tilde{\unicode[STIX]{x1D711}}}_{t}^{\unicode[STIX]{x1D6FF}}=v_{+}^{\unicode[STIX]{x1D6FF}}-v_{-}^{\unicode[STIX]{x1D6FF}}$ . A similar thing is true for the second derivatives $\ddot{\unicode[STIX]{x1D711}}_{t},\ddot{\tilde{\unicode[STIX]{x1D711}}}_{t}^{\unicode[STIX]{x1D6FF}}$ as well. As all the functions involved are quasi psh and bounded on $B_{\unicode[STIX]{x1D6FF}}$ , by Bedford–Taylor theory, positivity in (14) is preserved as we take the limit $\unicode[STIX]{x1D6FF}\rightarrow 0$ , and (14) holds restricted to $B_{\unicode[STIX]{x1D6FF}}$ , finishing the proof.◻

Proof. Fix $\unicode[STIX]{x1D700}>0$ for the moment. As it suffices to show convexity of $t\rightarrow \text{I}(\unicode[STIX]{x1D711}_{t})$ on $(\unicode[STIX]{x1D700},1-\unicode[STIX]{x1D700})$ , and $t\rightarrow \unicode[STIX]{x1D711}_{t}$ has minimal singularity potentials, without loss of generality we can assume that $t\rightarrow \unicode[STIX]{x1D711}_{t}$ is Lipschitz in $t$ and let $C>0$ be such that $|\unicode[STIX]{x1D711}_{t}-\unicode[STIX]{x1D711}_{t^{\prime }}|\leqslant C|t-t^{\prime }|$ . Let $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700},t}$ be the subgeodesics approximating $t\rightarrow \unicode[STIX]{x1D711}_{t}$ constructed above. Since $\text{I}(\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700},t})\rightarrow \text{I}(\unicode[STIX]{x1D711}_{t})$ as $\unicode[STIX]{x1D700}\rightarrow 0$ [Reference Berman and BoucksomBB10, Proposition 4.3], it suffices to prove the convexity of $t\rightarrow \text{I}(\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D700},t})$ . But this follows from Corollary 3.7.◻

Proof. By definition

$$\begin{eqnarray}V_{\unicode[STIX]{x1D6E9}}(x,z):=\sup \{U(x,z)\unicode[STIX]{x1D6E9}\text{-}\text{psh}:U\leqslant 0~\text{on}~M\}.\end{eqnarray}$$

Clearly $V_{\unicode[STIX]{x1D6E9}}(x,z)\geqslant V_{\unicode[STIX]{x1D703}}(x)$ since $V_{\unicode[STIX]{x1D703}}(x)$ is a candidate in the envelope. Moreover, we observe that for each $z\in \mathbb{C}\mathbb{P}^{1}$ , $V_{\unicode[STIX]{x1D6E9}}(x,z)$ is a $\unicode[STIX]{x1D703}$ -psh function and $V_{\unicode[STIX]{x1D6E9}}(x,z)\leqslant 0$ on $X$ , thus $V_{\unicode[STIX]{x1D6E9}}(x,z)\leqslant V_{\unicode[STIX]{x1D703}}(x)$ . Hence the conclusion.◻

Proof. As $H_{\infty }$ is pluripolar in $\mathbb{C}\mathbb{P}^{1}$ it suffices to prove (15) in $\mathbb{C}$ . As $F$ is $S^{1}$ -invariant it follows that $\unicode[STIX]{x1D6F7}(x,z)$ is also $S^{1}$ -invariant in $z$ . Using convolution as in § 3.2 we denote by $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}$ the approximants, i.e.

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}(x,z):=\int _{\mathbb{C}}\unicode[STIX]{x1D6F7}(x,\unicode[STIX]{x1D701})\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}(|z-\unicode[STIX]{x1D701}|^{2})\,dV(\unicode[STIX]{x1D701}),\end{eqnarray}$$

where $\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}$ is a family of smoothing kernels. For each $\unicode[STIX]{x1D700}>0$ , $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}$ is $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}$ -psh on $M$ where $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}:=\unicode[STIX]{x1D70B}_{1}^{\ast }\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D70B}_{2}^{\ast }\unicode[STIX]{x1D714}_{FS}$ , with $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D700}}>1$ decreasing to 1. Indeed,

$$\begin{eqnarray}dd_{x,z}^{c}\unicode[STIX]{x1D6F7}\star \unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}=dd^{c}\unicode[STIX]{x1D6F7}\star \unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}\geqslant (-\unicode[STIX]{x1D703}-\unicode[STIX]{x1D714}_{FS})\star \unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}\geqslant -\unicode[STIX]{x1D703}-\unicode[STIX]{x1D714}_{FS}\star \unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}.\end{eqnarray}$$

Moreover, since $\unicode[STIX]{x1D714}_{FS}$ is smooth on $\mathbb{C}\mathbb{P}^{1}$ , we have

$$\begin{eqnarray}|\unicode[STIX]{x1D714}_{FS}-\unicode[STIX]{x1D714}_{FS}\star \unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}|\leqslant \int _{\mathbb{C}}|\unicode[STIX]{x1D714}_{FS}(z)-\unicode[STIX]{x1D714}_{FS}(z-\unicode[STIX]{x1D701})|\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}(|\unicode[STIX]{x1D701}|^{2})\,dV(\unicode[STIX]{x1D701})\leqslant C\int _{\mathbb{ C}}|\unicode[STIX]{x1D701}|\,\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}(|z^{\prime }|^{2})\,dV(z^{\prime })\leqslant C\unicode[STIX]{x1D700}.\end{eqnarray}$$

This means that we can find $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D700}}$ decreasing to 1 such that $\unicode[STIX]{x1D714}_{FS}\star \unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}\leqslant \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D714}_{FS}$ . It can be shown in the same way as in § 3.2 that $G_{\unicode[STIX]{x1D700}}(z):=\text{I}(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}(\cdot ,z))$ is smooth and its complex Hessian is given by

(16) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D714}_{FS}+dd_{z}^{c}G_{\unicode[STIX]{x1D700}}(z)=(\unicode[STIX]{x1D70B}_{2})_{\star }(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}+dd_{x,z}^{c}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}})^{n+1}.\end{eqnarray}$$

To prove this we first explain how to compute all the second-order partial derivatives of $G_{\unicode[STIX]{x1D700}}$ . Fix $\unicode[STIX]{x1D709}\in \mathbb{C}=\mathbb{R}^{2}$ a unit vector and denote by $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}f(z):=\lim _{h\rightarrow 0^{+}}(f(z+h\unicode[STIX]{x1D709})-f(z))/h$ the derivative of $f$ in the direction  $\unicode[STIX]{x1D709}$ . Note that

$$\begin{eqnarray}\frac{1}{h}[\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}(|z+h\unicode[STIX]{x1D709}|^{2})-\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}(|z|^{2})]\rightarrow \unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D700}}(|z|^{2}))\end{eqnarray}$$

as $h\rightarrow 0$ uniformly in $z$ . It follows that

$$\begin{eqnarray}\frac{1}{h}[\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}(x,z+h\unicode[STIX]{x1D709})-\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}(x,z)]\rightarrow \unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}(z)\end{eqnarray}$$

as $h\rightarrow 0$ uniformly in $x,z$ . By the same arguments as in § 3.2 the first and second partial derivatives of $z\mapsto \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}(x,z)$ can be written as $C_{\unicode[STIX]{x1D700}}(u^{+}-u^{-})$ where $u^{\pm }$ are $\unicode[STIX]{x1D703}$ -psh functions with minimal singularities. They are thus uniformly bounded by a constant $C_{\unicode[STIX]{x1D700}}^{\prime }$ (which blows up as $\unicode[STIX]{x1D700}\rightarrow 0$ ). Thus the same arguments as in § 3.2 show that $z\mapsto G_{\unicode[STIX]{x1D700}}(z)$ is twice differentiable (even smooth). Set $g_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D700}}\log (1+|z|^{2})$ , where $g=\log (1+|z|^{2})$ is the potential of $\unicode[STIX]{x1D714}_{FS}$ in $\mathbb{C}$ . The second derivative $\unicode[STIX]{x2202}_{z}\unicode[STIX]{x2202}_{\bar{z}}$ of $G_{\unicode[STIX]{x1D700}}+g_{\unicode[STIX]{x1D700}}$ is given by

$$\begin{eqnarray}\unicode[STIX]{x2202}_{z}\unicode[STIX]{x2202}_{\bar{z}}(G_{\unicode[STIX]{x1D700}}+g_{\unicode[STIX]{x1D700}})=\int _{X}\unicode[STIX]{x2202}_{z}\unicode[STIX]{x2202}_{\bar{z}}(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}+g_{\unicode[STIX]{x1D700}})(\unicode[STIX]{x1D703}+dd_{x}^{c}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}})^{n}-n\,d_{x}\unicode[STIX]{x2202}_{z}(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}+g_{\unicode[STIX]{x1D700}})\wedge d_{x}^{c}\unicode[STIX]{x2202}_{\bar{z}}(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}+g_{\unicode[STIX]{x1D700}})\wedge (\unicode[STIX]{x1D703}+dd_{x}^{c}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}})^{n-1}.\end{eqnarray}$$

Let $H(z)$ denote the integrand in the right-hand side which is a positive measure on $X$ . Indeed, since it is a local property we can argue locally and use an approximation technique as in the proof of Corollary 3.7. Moreover, we infer that $(\sqrt{-1}/\unicode[STIX]{x1D70B})H(z)\,dz\wedge d\bar{z}$ is the Monge–Ampère measure of $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}+g_{\unicode[STIX]{x1D700}}$ with respect to the form $\unicode[STIX]{x1D70B}_{1}^{\ast }\unicode[STIX]{x1D703}$ , i.e. $(\unicode[STIX]{x1D70B}_{1}^{\ast }\unicode[STIX]{x1D703}+dd^{c}(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}+g_{\unicode[STIX]{x1D700}}))^{n+1}$ . This together with

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D714}_{FS}+dd_{z}^{c}G_{\unicode[STIX]{x1D700}}(z)=dd_{z}^{c}(g_{\unicode[STIX]{x1D700}}+G_{\unicode[STIX]{x1D700}})=dd_{z}^{c}\text{I}(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}(\cdot ,z)+g_{\unicode[STIX]{x1D700}}(z))\end{eqnarray}$$

justify the formula (16).

Fix $\unicode[STIX]{x1D712}:\mathbb{C}\rightarrow \mathbb{R}$ a smooth function with compact support in  $\mathbb{C}$ . We want to prove that

$$\begin{eqnarray}\int _{\mathbb{C}}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D714}_{FS}+dd^{c}G)=\int _{M}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6E9}+dd^{c}\unicode[STIX]{x1D6F7})^{n+1}.\end{eqnarray}$$

The above formula is true for the approximants $G_{\unicode[STIX]{x1D700}}$ and $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}$ as we discussed above. Now we explain how we can insure the convergence when we take the limit in (16) as $\unicode[STIX]{x1D700}\rightarrow 0$ . To deal with the left-hand side we prove that $G_{\unicode[STIX]{x1D700}}$ converge pointwise to  $G$ . For fixed $z\in \mathbb{C}$ we can find constants $c_{\unicode[STIX]{x1D700}}$ converging to $0$ such that $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}}+c_{\unicode[STIX]{x1D700}}$ decreases to  $\unicode[STIX]{x1D6F7}$ . Then $G_{\unicode[STIX]{x1D700}}(z)$ converges to $G(z)$ by basic properties of the $\text{I}$ functional [Reference Berman and BoucksomBB10, Proposition 4.3]. As $G_{\unicode[STIX]{x1D700}}$ is uniformly bounded independent of $\unicode[STIX]{x1D700}$ , the convergence of the current $dd^{c}G_{\unicode[STIX]{x1D700}}$ follows. Now we treat the convergence of the right-hand side of (16). Fix a Kähler form $\unicode[STIX]{x1D714}_{M}$ on $M$ and $\unicode[STIX]{x1D6FF}>0$ . Let $U$ be an open neighborhood of the pluripolar set $E=(X\setminus \operatorname{Amp}(\{\unicode[STIX]{x1D703}\}))\times H_{\infty }$ such that $\operatorname{Cap}_{\unicode[STIX]{x1D714}_{M}}(U)<\unicode[STIX]{x1D6FF}$ . Note that Theorem 2.8 gives that

$$\begin{eqnarray}\operatorname{Cap}_{\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}}(U)\leqslant C\operatorname{Cap}_{\unicode[STIX]{x1D714}_{M}}(U)^{1/n}=O(\unicode[STIX]{x1D6FF}^{1/n}),\end{eqnarray}$$

where $C$ is independent of $\unicode[STIX]{x1D700}$ . And, by Bedford–Taylor theory [Reference Bedford and TaylorBT82] the Monge–Ampère measure $(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D700}}+dd^{c}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D700}})^{n+1}$ converges to $(\unicode[STIX]{x1D6E9}+dd^{c}\unicode[STIX]{x1D6F7})^{n+1}$ in $M\setminus U$ . Letting $\unicode[STIX]{x1D700}\rightarrow 0$ and then $\unicode[STIX]{x1D6FF}\rightarrow 0$ we arrive at the conclusion.◻

Proof. When restricted to $X\times D$ , we have $\unicode[STIX]{x1D6F7}+\unicode[STIX]{x1D70C}$ is $\unicode[STIX]{x1D70B}_{1}^{\ast }\unicode[STIX]{x1D703}$ -psh and has boundary values $\unicode[STIX]{x1D711}_{0,1}$ , thus by definition, $\unicode[STIX]{x1D6F7}\leqslant \unicode[STIX]{x1D711}_{t}-\unicode[STIX]{x1D70C}<F$ . It follows from [Reference Bedford and TaylorBT82] that $(\unicode[STIX]{x1D6E9}+dd^{c}\unicode[STIX]{x1D6F7})^{n+1}=0$ in $X\times D$ , which in turn implies that $\text{I}(\unicode[STIX]{x1D6F7}(\cdot ,z)+\unicode[STIX]{x1D70C}(z))$ is harmonic in $D$ thanks to Lemma 3.10.◻

Proof. Fix $f_{0},f_{1}$ two smooth functions in $X$ and denote by $\unicode[STIX]{x1D711}_{i}=P_{\unicode[STIX]{x1D703}}(f_{i}),i=0,1$ the envelopes of $f_{0},f_{1}$ respectively. Let $\unicode[STIX]{x1D711}_{t}$ be the geodesic connecting $\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$ . Observe also that by approximating any two given potentials with minimal singularities with a sequence of smooth functions, it suffices to prove linearity of $\text{I}$ along  $\unicode[STIX]{x1D711}_{t}$ .

Let $F$ be a function on $M$ which is $S^{1}$ -invariant in $X\times \mathbb{C}$ in the variable $z$ and such that $F=f_{0,1}$ on $X\times \unicode[STIX]{x2202}D$ and $+\infty$ elsewhere. Consider a sequence $(F_{j})_{j}$ of smooth functions $F_{j}\uparrow F$ , which are also $S^{1}$ -invariant in $X\times \mathbb{C}$ in the variable $z$ and such that $F_{j}+\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D711}_{\log |z|}(x)$ in $X\times D$ . Let $\unicode[STIX]{x1D6F7}_{j}$ be the envelope on $M$ of $F_{j}$ with respect to $\unicode[STIX]{x1D6E9}$ . Then $(\unicode[STIX]{x1D6E9}+dd^{c}\unicode[STIX]{x1D6F7}_{j})^{n+1}$ is supported on $\{\unicode[STIX]{x1D6F7}_{j}=F_{j}\}$ . As $X\times \unicode[STIX]{x2202}D$ is non-pluripolar in $M$ it follows that $\unicode[STIX]{x1D6F7}_{j}$ is uniformly bounded from above. Thus $\unicode[STIX]{x1D6F7}_{j}$ converges increasingly (almost everywhere) to $\unicode[STIX]{x1D6F7}$ a $\unicode[STIX]{x1D6E9}$ -psh function with minimal singularities. We claim that

(17) $$\begin{eqnarray}\int _{\{\unicode[STIX]{x1D6F7}<P_{\unicode[STIX]{x1D6E9}}(F)\}}(\unicode[STIX]{x1D6E9}+dd^{c}\unicode[STIX]{x1D6F7})^{n+1}=0.\end{eqnarray}$$

To see this we observe that the Monge–Ampère measure of $\unicode[STIX]{x1D6F7}$ is concentrated on $X\times \unicode[STIX]{x2202}D$ and that $\{\unicode[STIX]{x1D6F7}<P_{\unicode[STIX]{x1D6E9}}(F)\}\subset X\times (\mathbb{C}\mathbb{P}^{1}\setminus \unicode[STIX]{x2202}D)$ since $\unicode[STIX]{x1D6F7}=P_{\unicode[STIX]{x1D6E9}}(F)=\unicode[STIX]{x1D711}_{0,1}$ on $X\times \unicode[STIX]{x2202}D$ . Indeed, on any open relatively compact subset $K$ of $X\times (\mathbb{C}\mathbb{P}^{1}\setminus \unicode[STIX]{x2202}D)$ one has that $\unicode[STIX]{x1D6F7}_{j}<F_{j}$ for $j$ large enough (since $F_{j}$ increases to $+\infty$ uniformly in $K$ and $\unicode[STIX]{x1D6F7}_{j}$ is uniformly bounded from above). By the continuity property of the complex Monge–Ampère operator we get that $(\unicode[STIX]{x1D6E9}+dd^{c}\unicode[STIX]{x1D6F7})^{n+1}(K)=0$ . It follows from (17) and the domination principle [Reference Boucksom, Eyssidieux, Guedj and ZeriahiBEGZ10, Corollary 2.5] that $\unicode[STIX]{x1D6F7}=P_{\unicode[STIX]{x1D6E9}}(F)$ .

Now, we claim that $\unicode[STIX]{x1D6F7}+\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D711}_{\log |z|}$ in $X\times D$ . Indeed, consider

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}U_{0}(x,z):=\unicode[STIX]{x1D711}_{0}(x)+A(\log (|z|^{2}+3)-\log (|z|^{2}+1)-\log 2);\quad \\ U_{1}(x,z):=\unicode[STIX]{x1D711}_{1}(x)+A(\log |z|^{2}-\log (|z|^{2}+1)+\log (e^{2}+1)-2).\quad \end{array}\right.\end{eqnarray}$$

For $A>0$ big enough $U:=\max (U_{0},U_{1})=\unicode[STIX]{x1D711}_{0,1}$ on $\unicode[STIX]{x2202}D$ and it is $\unicode[STIX]{x1D6E9}_{A}$ -psh, where $\unicode[STIX]{x1D703}_{A}:=\unicode[STIX]{x1D70B}_{1}^{\ast }\unicode[STIX]{x1D703}+A\unicode[STIX]{x1D70B}_{2}^{\ast }\unicode[STIX]{x1D714}_{FS}$ . So we can apply our previous analysis with this $(1,1)$ -form $\unicode[STIX]{x1D6E9}_{A}$ instead of $\unicode[STIX]{x1D6E9}$ . By definition of envelope we have $\unicode[STIX]{x1D6F7}\geqslant U$ and in particular $\unicode[STIX]{x1D6F7}\geqslant \unicode[STIX]{x1D711}_{0,1}$ on $X\times \unicode[STIX]{x2202}D$ . Moreover, for each $z\in \unicode[STIX]{x2202}D$ , $\unicode[STIX]{x1D6F7}(\cdot ,z)$ is $\unicode[STIX]{x1D703}$ -psh and dominated by $F=f_{0,1}$ . It then follows that $\unicode[STIX]{x1D6F7}\leqslant \unicode[STIX]{x1D711}_{0,1}$ on $X\times \unicode[STIX]{x2202}D$ , giving the equality on the boundary. Furthermore, it follows from the proof of Lemma 3.11 and the continuity of the Monge–Ampère operator that the Monge–Ampère measure $(\unicode[STIX]{x1D70B}_{1}^{\ast }\unicode[STIX]{x1D703}+dd^{c}\unicode[STIX]{x1D70C}+dd^{c}\unicode[STIX]{x1D6F7})^{n+1}$ vanishes in $X\times D$ . Proposition 3.3 thus insures that $\unicode[STIX]{x1D6F7}+\unicode[STIX]{x1D70C}$ is the unique weak geodesic with minimal singularities joining $\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D711}_{1}$ . Hence the claim.

Now, thanks to Lemma 3.11 and [Reference Berman and BoucksomBB10, Proposition 4.3], we know that $\text{I}(\unicode[STIX]{x1D6F7}(\cdot ,z)+\unicode[STIX]{x1D70C}(z))=\text{I}(\unicode[STIX]{x1D711}_{\log |z|})$ is harmonic in $D$ (and $S^{1}$ -invariant!). Hence, $\text{I}$ is linear along $\unicode[STIX]{x1D711}_{t}$ , with $t=\log |z|$ . This is what we wanted.◻

Proof. As all complexifications $U^{l}\in \operatorname{PSH}(X\times D_{l},\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703})$ satisfy the appropriate complex Monge–Ampère equation on the domains $\operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D_{l}$ and are locally bounded there, it follows from continuity property of the complex Monge–Ampère operator that the complexification of $t\rightarrow v(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})_{t}$ satisfies the homogeneous Monge–Ampère equation on $\operatorname{Amp}(\{\unicode[STIX]{x1D703}\})\times D_{\infty }$ as well. Since all the curves $t\rightarrow u_{t}^{l}$ are uniformly $t$ -Lipschitz continuous, so is their limit $t\rightarrow v(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})_{t}$ , hence Proposition 3.3 gives that $t\rightarrow v(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})_{t}$ must be a weak geodesic ray, i.e., for any closed interval $I\subset [0,\infty )$ , the restriction $I\ni t\rightarrow v(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})_{t}\in \operatorname{PSH}(X,\unicode[STIX]{x1D703})$ is the weak geodesic joining the potentials corresponding to the endpoints of  $I$ .◻

Proof. The proof is an adaptation of the arguments of [Reference DarvasDar17b, Theorem 2.5] to our more general setting. As $c_{\unicode[STIX]{x1D713}}$ only depends on $\unicode[STIX]{x1D713}$ it is enough to work with the special subgeodesic ray $t\rightarrow \unicode[STIX]{x1D713}_{t}:=\max (V_{\unicode[STIX]{x1D703}}-t,\unicode[STIX]{x1D713})$ . The cocycle formula (19) implies that $c_{\unicode[STIX]{x1D713}}=c_{\unicode[STIX]{x1D713}+c}$ , thus we can assume that $\unicode[STIX]{x1D713}\leqslant V_{\unicode[STIX]{x1D703}}\leqslant 0$ . By [Reference Berman, Boucksom, Guedj and ZeriahiBBGZ13, Proposition 2.8]

$$\begin{eqnarray}\int _{X}(\unicode[STIX]{x1D713}_{t}-V_{\unicode[STIX]{x1D703}})\,\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}\leqslant \text{I}(\unicode[STIX]{x1D713}_{t})\leqslant (n+1)^{-1}\int _{X}(\unicode[STIX]{x1D713}_{t}-V_{\unicode[STIX]{x1D703}})\,\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}\end{eqnarray}$$

so our claim is equivalent to showing that

$$\begin{eqnarray}\unicode[STIX]{x1D713}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703})\Longleftrightarrow t^{-1}\int _{X}(\unicode[STIX]{x1D713}_{t}-V_{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}\rightarrow 0.\end{eqnarray}$$

Fix $s\in (0,1)$ . Note that on $\{\unicode[STIX]{x1D713}>V_{\unicode[STIX]{x1D703}}-t\}$ , the two measures $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}$ and $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}}^{n}$ coincide. Additionally, using that $X=\{\unicode[STIX]{x1D713}\leqslant V_{\unicode[STIX]{x1D703}}-t\}\cup \{V_{\unicode[STIX]{x1D703}}-t<\unicode[STIX]{x1D713}\leqslant V_{\unicode[STIX]{x1D703}}-st\}\cup \{V_{\unicode[STIX]{x1D703}}-st<\unicode[STIX]{x1D713}\}$ we can estimate the right-hand side above as follows:

$$\begin{eqnarray}0\geqslant \int _{X}\frac{\unicode[STIX]{x1D713}_{t}-V_{\unicode[STIX]{x1D703}}}{t}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}\geqslant -\int _{\{\unicode[STIX]{x1D713}\leqslant V_{\unicode[STIX]{x1D703}}-t\}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}-\int _{\{\unicode[STIX]{x1D713}\leqslant V_{\unicode[STIX]{x1D703}}-ts\}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}}^{n}-s\operatorname{Vol}(\unicode[STIX]{x1D703}).\end{eqnarray}$$

By [Reference Guedj and ZeriahiGZ07, Lemma 1.2], $\unicode[STIX]{x1D713}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703})$ if and only if $\int _{\{\unicode[STIX]{x1D713}\leqslant V_{\unicode[STIX]{x1D703}}-t\}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D713}_{t}}^{n}\rightarrow 0$ as $t\rightarrow +\infty$ . Hence, the above estimates give the conclusion.◻

Proof. We go back to the construction of $t\rightarrow v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{t}$ in (18). By Theorem 3.12, for each $l$ fixed, the function $t\rightarrow \text{I}(u_{t}^{l})$ is linear, and hence we can write

$$\begin{eqnarray}\text{I}(u_{t}^{l})=\text{I}(V_{\unicode[STIX]{x1D703}})+\frac{t}{l}(\text{I}(\max (\unicode[STIX]{x1D713},V_{\unicode[STIX]{x1D703}}-l))-\text{I}(V_{\unicode[STIX]{x1D703}}))=\frac{t}{l}(\text{I}(\max (\unicode[STIX]{x1D713},V_{\unicode[STIX]{x1D703}}-l))-\text{I}(V_{\unicode[STIX]{x1D703}})).\end{eqnarray}$$

Letting $l\rightarrow +\infty$ , by [Reference Berman and BoucksomBB10, Proposition 3.3] we obtain (20). If $t\rightarrow v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{t}$ is constant equal to $V_{\unicode[STIX]{x1D703}}$ , it follows that $c_{\unicode[STIX]{x1D713}}=0$ , thus by Proposition 3.14 we get $\unicode[STIX]{x1D713}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703})$ . Conversely, if $\unicode[STIX]{x1D713}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703})$ , then Proposition 3.14 yields that $c_{\unicode[STIX]{x1D713}}=0$ , hence $\text{I}$ is constant along $v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})$ , and thus $v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})$ is constant.◻

Proof. One can repeat the argument in [Reference DarvasDar17b, Theorem 5.3]. Fix $\unicode[STIX]{x1D70F}\in \mathbb{R}$ . The fact that $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }$ is $\unicode[STIX]{x1D703}$ -psh follows from Kiselman’s minimum principle [Reference KiselmanKis78]. Suppose that $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }\neq -\infty$ and fix $C>0$ . Since $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }\leqslant \unicode[STIX]{x1D719}_{0}$ , it results that $P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }+C,\unicode[STIX]{x1D719}_{0})\geqslant \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }$ . Hence we only have to argue that

$$\begin{eqnarray}P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }+C,\unicode[STIX]{x1D719}_{0})\leqslant \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }.\end{eqnarray}$$

Let $[0,1]\ni t\rightarrow g_{t}^{l},h_{t}\in \operatorname{PSH}(X,\unicode[STIX]{x1D703})$ , $l\geqslant 0$ , be the weak geodesic segments defined by the formulas

$$\begin{eqnarray}\displaystyle & g_{t}^{l}=\unicode[STIX]{x1D719}_{tl}-tl\unicode[STIX]{x1D70F}, & \displaystyle \nonumber\\ \displaystyle & h_{t}=P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }+C,\unicode[STIX]{x1D719}_{0})-Ct. & \displaystyle \nonumber\end{eqnarray}$$

Then we have $h_{0}\leqslant \unicode[STIX]{x1D719}_{0}=\lim _{t\rightarrow 0}g_{t}^{l}=g_{0}^{l}$ and $h_{1}\leqslant \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }\leqslant g_{1}^{l}$ for any $l\geqslant 0$ . Hence, by definition of weak geodesics (8) we have

$$\begin{eqnarray}h_{t}\leqslant g_{t}^{l},\quad t\in [0,1],l\geqslant 0.\end{eqnarray}$$

Taking the infimum in the above estimate over $l\in [0,+\infty )$ and then taking the supremum over $t\in [0,1]$ , we obtain

$$\begin{eqnarray}P_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }+C,\unicode[STIX]{x1D719}_{0})\leqslant \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D70F}}^{\ast }.\end{eqnarray}$$

Letting $C\rightarrow +\infty$ we obtain the last statement of the proposition.◻

Proof. We can assume without loss of generality that $\unicode[STIX]{x1D713}\leqslant 0$ . The implication $(\text{i})\Longrightarrow (\text{iii})$ follows from Theorem 2.15 while the implications $(\text{iii})\Longrightarrow (\text{ii})\Longrightarrow (\text{iv})$ and $\text{(iii)}\Longrightarrow (\text{v})$ are trivial. The implication $(\text{iv})\Longrightarrow (\text{v})$ simply follows from Proposition 2.14 and the domination principle (Proposition 2.4).

We now prove that $(\text{v})\Longrightarrow (\text{i})$ . Suppose that $P_{[\unicode[STIX]{x1D703},\unicode[STIX]{x1D713}]}(V_{\unicode[STIX]{x1D703}})=V_{\unicode[STIX]{x1D703}}$ . From the construction of the ray $t\rightarrow v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{t}$ in (18) it automatically follows that $v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{t}\geqslant \unicode[STIX]{x1D713}$ . This trivially gives $v_{0}^{\ast }=\inf _{t\in [0,\infty )}v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{t}\geqslant \unicode[STIX]{x1D713}$ . By definition of envelope we have $V_{\unicode[STIX]{x1D703}}\geqslant P_{[\unicode[STIX]{x1D703},v_{0}^{\ast }]}(V_{\unicode[STIX]{x1D703}})\geqslant P_{[\unicode[STIX]{x1D703},\unicode[STIX]{x1D713}]}(V_{\unicode[STIX]{x1D703}})=V_{\unicode[STIX]{x1D703}}$ . Combining this with Lemma 3.17 we obtain that $v_{0}^{\ast }=P_{[\unicode[STIX]{x1D703},v_{0}^{\ast }]}(V_{\unicode[STIX]{x1D703}})=V_{\unicode[STIX]{x1D703}}$ . As the ray $t\rightarrow v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{t}$ is decreasing in $t$ , this automatically gives that $V_{\unicode[STIX]{x1D703}}=v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{0}\geqslant v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{t}\geqslant v_{0}^{\ast }=V_{\unicode[STIX]{x1D703}}$ , hence $t\rightarrow v(V_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D713})_{t}$ is constant. Invoking Lemma 3.15 we obtain that $\unicode[STIX]{x1D713}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703})$ .◻

Proof. Since $V_{\unicode[STIX]{x1D703}_{j}}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703}_{j}),j=1,2$ , the implication $(\text{ii})\Longrightarrow (\text{i})$ is trivial. Assume (i) holds. The implication $(\Longrightarrow )$ in (ii) follows from [Reference Di NezzaDiN15, Theorem B]. Assume that $\unicode[STIX]{x1D711}_{j}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703}_{j}),j=1,2$ . We want to show that $\unicode[STIX]{x1D711}:=\unicode[STIX]{x1D711}_{1}+\unicode[STIX]{x1D711}_{2}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2})$ . By assumption that (i) holds we get that $V_{\unicode[STIX]{x1D703}_{1}}+V_{\unicode[STIX]{x1D703}_{2}}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2})$ . Hence, by definition of envelopes we can write

$$\begin{eqnarray}P_{[\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2},\unicode[STIX]{x1D711}]}(V_{\unicode[STIX]{x1D703}_{1}}+V_{\unicode[STIX]{x1D703}_{2}})\geqslant P_{[\unicode[STIX]{x1D703}_{1},u]}(V_{\unicode[STIX]{x1D703}_{1}})+P_{[\unicode[STIX]{x1D703}_{2},v]}(V_{\unicode[STIX]{x1D703}_{2}})=V_{\unicode[STIX]{x1D703}_{1}}+V_{\unicode[STIX]{x1D703}_{2}}.\end{eqnarray}$$

Ultimately, it follows from Theorem 1.2 that $\unicode[STIX]{x1D711}\in {\mathcal{E}}(X,\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2})$ .◻