PomeCorollary 14.23.7.3DF5F7F34405 Let $n\in \{ 2,3\} $, and let $u,v\in C^2(X)$ satisfy $A,B\geq 0$ and $A^ n=B^ n=f\, \mathrm{d}V$, where $f\geq 0$ is continuous and positive on a dense set. If $n=3$, assume (14.23.2); for instance, $\mathrm{dd}^{\mathrm{c}}\omega =0$ suffices. Then $u-v$ is constant.
Corollary 14.23.73DF5F7F34405
Proof. Put $D=A-B$ and $C=(A+B)/2$. At a point where the common top power vanishes, the algebraic expressions in the proof of 14.23.1 are still nonpositive: in dimension two $D^2=-2A\wedge B$, and in dimension three $2C\wedge D^2=-A^2\wedge B-A\wedge B^2$. The same integral identities force these continuous expressions to vanish everywhere. The equality cases give $D=0$ on $\{ f>0\} $, and density and continuity then give $D=0$ on $X$. As in the proof of the theorem, $u-v$ is then constant. $\square$
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