Corollary 14.24.17527FB17149BB

Proof. The functions $u,v$ are continuous by corollary 14.19.29, and in dimension one the assertion follows from $\mathrm{dd}^{\mathrm{c}}(u-v)=0$ and the maximum principle. Let $n\geq 2$, suppose that $w=u-v$ is nonconstant, and put $K=\{ w=\min _ Xw\} $. The proof of lemma 14.24.11 applies with $U=X$, since proposition 14.24.5 gives the pushforward identity and local coercivity on every chart. Thus $K$ has zero volume.

Fix $B_0>\| 1\| _{L^2(\omega ^ n)}$ and the corresponding $\varepsilon $ in proposition 14.24.13. Choose $A>1$ with $\varepsilon A>C_0$, and a smooth cutoff function $\chi $, equal to one on a neighborhood $W$ of $K$ and supported in an open set of volume so small that $g=1+(A-1)\chi $ satisfies $\| g\| _2<B_0$. The resulting test potential $q$ has density $G\geq \varepsilon g$. Choose $\kappa <1$ with $C_0<\kappa \varepsilon A$. Then $F<\kappa G$ almost everywhere on $W$. By continuity of $w$, a sufficiently thin set $\{ w<\min _ Xw+\eta \} $ lies inside $W$, which contradicts proposition 14.24.2. Hence $w$ is constant. $\square$