Lemma 15.27.2366F6832AB46

Proof. We first show that the scalars $c_ h(q_ j)$ lie in a compact subset of $(0,+\infty )$. Fix a Gauduchon metric $\gamma $ on $X$ (definition 11.4.1; in dimension one, any Hermitian metric). Pairing the equation for $q_ j$ with $\gamma ^{n-1}$, as in proposition 15.14.2 with $\theta =h$ and barrier $\rho =0$, gives

Since $q_ j^{1/n}\to q^{1/n}$ in $L^ n$, the integrals on the left converge to the corresponding integral for $q$, which is positive. Hence the scalars are bounded above. The lower-density estimate of theorem 14.19.19 gives $c_ h(q_ j)\| q_ j\| _{L^ p}\geq c_ p>0$, so they are also bounded away from zero.

Consequently, the normalized continuous solutions $u_ j$ of $\mathrm{MA}_ h(u_ j)=c_ h(q_ j)q_ j\, \mathrm{d}V$ have uniformly bounded $L^ p$ densities. By theorem 14.19.21, every subsequence has a further subsequence along which $u_ j$ converges uniformly to some $u$ and $c_ h(q_ j)$ converges to some $c>0$. Continuity of bounded Monge–Ampère products under uniform convergence gives $\mathrm{MA}_ h(u)=cq\, \mathrm{d}V$, and uniqueness of the scalar gives $c=c_ h(q)$. Since every subsequence of $(c_ h(q_ j))$ has a further subsequence converging to $c_ h(q)$, the whole sequence converges to $c_ h(q)$. $\square$