Corollary 14.22.15`EAC3CF41BD99`

Corollary 14.22.15.EAC3CF41BD99 Assume $n=2$ and $\mathrm{dd}^{\mathrm{c}}\omega =0$. Let $f\geq 0$ be smooth, with a zero set of empty interior, and let $c>0$. Then any two smooth $\omega $-plurisubharmonic solutions of

\[ (\omega +\mathrm{dd}^{\mathrm{c}}u)^2=cf\, \mathrm{d}V \]

normalized by $\sup _ Xu=0$ coincide.

Proof. Here $T=A+B$ and $\mathrm{dd}^{\mathrm{c}}T=2\mathrm{dd}^{\mathrm{c}}\omega =0$. The set $\{ f>0\} $ is dense, so the proposition, applied to the density $cf$, makes the difference of the two solutions constant, and the normalization makes this constant zero. $\square$

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BibTeX citation

@misc{pome-tag-EAC3CF41BD99,
  author = {{Pome contributors}},
  title = {Corollary 14.22.15},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/EAC3CF41BD99/}
}

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Corollary 14.22.15EAC3CF41BD99

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Corollary 14.22.15`EAC3CF41BD99`