Lemma 15.7.15C54B7FA3A8B

Proof. Extend $\varphi $ to a smooth function $\widetilde\varphi $ on $\overline B$. For sufficiently large $A$, the function $\underline v=\widetilde\varphi +A(|z|^2-R^2)$ is a smooth strict subsolution with the prescribed boundary values: $\theta +\mathrm{dd}^{\mathrm{c}}\underline v$ is positive and its determinant is at least $(\max _{\overline B}F)\beta ^ n$. For $n\geq 2$, the classical smooth Dirichlet theorem of Guan–Li therefore applies on the ball, with the arbitrary smooth background $\chi =\theta $ and the smooth positive right-hand side $F$. Their operator is $(\mathrm{i}/2)\partial \bar\partial $, so our potential is $\pi $ times theirs. No closedness condition is needed here; such conditions enter only in their separate theorems on compact manifolds. For $n=1$ the lemma is the Dirichlet problem for the Poisson equation. For uniqueness, integrate the derivative of $\log \det $ along the segment between the two positive matrices: the difference of the logarithms of the two determinants becomes a uniformly elliptic linear operator with smooth coefficients applied to the difference of the solutions, and the maximum principle applies. $\square$