Lemma 18.42.13FBB9C424D045

Proof. Take $b_ R=|z|^2/R^2$ on a disk containing $D_ t$, and cut it off before the radius $3R$, so that its negative curvature is supported outside $D_ t$. This function is uniformly bounded and its Hessian is $O(R^{-2})$. On the support of its negative curvature, the curvature $m\rho $ is comparable to $q/R^2$; since $q\to \infty $, it absorbs the negative part. Inside $D_ t$, the function $b_ R$ contributes the curvature $c/R^2$. Elsewhere in a fixed coordinate disk, the original curvature is bounded below by a fixed multiple of $m|z|^{2r-2}$, and off that disk it is at least $cm$. Since $mR^2\to \infty $, all these bounds dominate $c/R^2$ and the bounded Ricci term of $\nu $.

We use the $L^2$ estimate for $\bar\partial $ with a pointwise lower bound for the curvature of a singular metric, in the form described in section 25.16. As in the proof of theorem 18.42.1, we regard $L^ m$-valued $(0,1)$-forms as $(1,1)$-forms with values in $L^ m\otimes K_ C^{-1}$ and sections of $L^ m$ as $(1,0)$-forms with values in the same bundle; this identification preserves the pointwise norms and brings in the Ricci term of $\nu $. The estimate then reads as follows: if $H$ is a singular metric on $L^ m$ and the curvature of $L^ m\otimes K_ C^{-1}$ is at least $\lambda \nu $ for a continuous function $\lambda >0$, then

On the support of $\alpha $ we have $\lambda ^{-1}=O(R^2/q)$, $\phi _ t=\phi $, and $b_ R$ is bounded. Elsewhere $\phi _ t\leq \phi $ gives $\mathrm{e}^{-m\phi }\leq C\mathrm{e}^{-\psi _ m}$. These comparisons yield (18.42.14). Choosing the solution of minimal norm in the singular weighted Hilbert space makes the correction linear in its source.

The estimate requires a continuous lower bound $\lambda $. We take $\lambda $ equal to $c/R^2$ inside $D_ t$ and to $cq/R^2$ on the exterior annulus, interpolating across a thin exterior collar of $\Gamma _ t$. For the cutoff sources used below, whose supports are separated from $\Gamma _ t$, the collar can be chosen to miss the source. For an exterior source whose support accumulates on $\Gamma _ t$, we let the collar shrink to $\Gamma _ t$; since $\Gamma _ t$ has zero area, dominated convergence in the integrals of $\lambda ^{-1}|\alpha |^2$ and weak compactness in the fixed weighted Hilbert space give the same estimate. The singular metric is handled by approximation, which gives the same estimate for a weak solution. Where the source vanishes, the Cauchy–Riemann equation in the sense of distributions makes the solution holomorphic. Near $x$, the weight is $|z|^{-2q}$ times a smooth positive factor. A holomorphic term $z^ j$ is square integrable against it precisely when $j\geq q$, so the solution has the required vanishing order. $\square$