18.42.26 Growth ranges and compact examplesE7C597B62D6D

Proof. By lemma 18.42.10, $c_ R-1=O(R)$ in every fixed norm required here. The observation points in (18.42.28) satisfy

Since the slope of the boundary graph tends to zero, the argument of $\Phi $ in theorem 18.42.15 is $\tau +o(1)$ whenever $\sqrt q R\to 0$, and this condition is equivalent to $q=o(m^{1/(r+1)})$. Moreover, $a(z)=1+o(1)$ and $\rho (z)=r^2|z|^{2r-2}(1+o(1))$ at these points, so the two normalizations agree. Thus theorem 18.42.15 proves the law for all data in this range.

The same reasoning shows that (18.42.29) is sufficient. For necessity, suppose that (18.42.29) fails. On a subsequence, choose an angle realizing the maximal boundary displacement and observe at $|z|=R$ with $\tau =0$. A nonzero finite subsequential limit of the scaled displacement gives a nonzero Gaussian argument, and an infinite one gives saturation to zero or one. Radial and normal distances have relative error tending to zero, because the graph has vanishing slope. In either case the limit differs from $1/2$.

It remains to construct the compact examples. On $\mathbb P^1$ choose a radial metric on $\mathcal O(d)$ whose local potential $f(u)$, $u=|w|^2$, equals $u^ r$ near zero, such that $g(u)=uf'(u)$ is strictly increasing on $(0,\infty )$ and tends to $d$, and such that $g(u)=d-\epsilon /u$ for large $u$, with a constant $\epsilon >0$. Such a smooth interpolation exists for a sufficiently large integer $d$: prescribe a positive function $g'$, equal to $r^2u^{r-1}$ near zero and to $\epsilon /u^2$ near infinity, with total integral $d$. Then $f=d\log u+\mathrm{constant}+\epsilon /u$ near infinity, so the potential in the frame at infinity is smooth with strictly positive curvature. Choose a smooth radial volume form equal to $\mathrm{d}A_ w$ near zero.

For sufficiently small $t$, the hole $D_ t$ lies in the region where $f(u)=u^ r$ and equals the disk $|w|<R$: rotational invariance and uniqueness make it a disk, and its curvature mass is $rR^{2r}=t$. Hence the transition and saturation statements of theorem 18.42.15 apply to this compact example.

Now use instead the coordinate $z$ defined by $w=z+cz^2$, with a constant $c>0$. In this coordinate the same compact metric satisfies $\phi (z)=|z|^{2r}+O(|z|^{2r+1})$, the volume form still has $a(0)=1$, and vanishing orders are unchanged. At the uncorrected observation point $z=R$ with $\tau =0$,

and the normalizing factors computed in the two coordinates have ratio tending to one. If $q$ is not $o(m^{1/(r+1)})$, there is a subsequence on which $\sqrt q R$ tends to a positive finite number or to infinity. Theorem 18.42.15, together with its saturation statement when the argument diverges, then gives a limit strictly larger than $1/2$. This proves that the range is sharp. $\square$