Proposition 12.21.458574FAEC5B7

Proof. In proposition 12.21.1 take $a=b>0$ and $r=0$. Fix a point $z_0\in E$ and a flat coordinate $\zeta =z-z_0$ on a small disk about $z_0$. Choose $q$ smooth, positive outside $z_0$, and equal to $|\zeta |^{2m}$ on that disk; a positive smooth extension exists away from the disk. For $m=1$, polar coordinates give

and (12.21.2) gives the asymptotic formula of the statement. If $m>1$, the substitution $s=|\zeta |^2=\epsilon ^{1/m}t$ gives

so the common order is $1-1/m$. In the logarithmic case, $\lambda _\mu (\theta )\geq 0$ by semipositivity, while $\lambda _\mu (\theta )\leq \lambda _\mu (\theta +\epsilon h)\to 0$ by monotonicity; hence $\mathcal V_\mu (\theta )=0$. Moreover, the quotient of the volume by $\epsilon ^\delta $ diverges for every $\delta >0$, so both orders vanish. Finally, $\mathrm{d}\theta =\mathrm{d} q\wedge \omega _2\ne 0$, so these backgrounds are not closed. $\square$