Lemma 11.9.1A592FEE54E20

Proof. Start with any metric $k^0$, and put $\mathcal Lu=(\mathrm{dd}^{\mathrm{c}}u\wedge \Theta )/\mu $. This scalar operator is uniformly elliptic, has no zeroth-order term, and its kernel consists of the constants by the maximum principle. On Hölder spaces, the homotopy to a Laplace operator used in the proof of theorem 11.4.2 shows that its index is zero, so its cokernel is one-dimensional. Since $\mathrm{dd}^{\mathrm{c}}\Theta =0$, integration by parts gives $\int _ X\mathcal Lu\, \mu =0$, and therefore its image consists exactly of the smooth functions of zero $\mu $-mean. Hence we can solve

whose right-hand side has integral zero, and elliptic regularity makes $u$ smooth. Then $k=k^0\mathrm e^{-u}$ has the required curvature. $\square$