We return to the threefold $X=H_3$ and its form $\omega $.
Averaging along the elliptic fibers removes the variation measured by (23.1.5). The Poincaré inequality on the fibers bounds the distance from a potential to its average by the excess mass. Bounded regularization preserves this inequality, which therefore identifies all bounded potentials of minimal mass and shows that potentials of almost minimal mass are close to them.
Let $E$ act on $X$ by scalar multiplication: $\tau _ a[z]=[\mathrm{e}^ a z]$ for $a\in E$. The action preserves $\alpha $, $\beta $ and $\omega $, and $\mathrm{d}A_ E$ is the Haar probability measure of $E$. For a bounded Borel function $u:X\to \mathbb R$, set
\[ \mathrm{d}\nu =\frac{\omega ^3}{6}=\frac12\alpha ^2\wedge \beta , \qquad Au(p)=\int _ Eu(\tau _ a(p))\, \mathrm{d}A_ E(a) \quad (p\in X). \]
Thus $Au$ is constant on each fiber, and $\nu $ is the probability measure whose disintegration along $\pi $ has fiber measures $\mathrm{d}A_ E$ and base measure $\omega _{\mathrm{FS}}^2$.
Proposition 23.1.38.E42306969D71 For every $u\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)$, the average $Au$ is the pullback of a bounded $\omega _{\mathrm{FS}}$-plurisubharmonic function on $\mathbb P^2$, and
18B65325175F
23.1.39
\begin{equation} \| u-Au\| _{L^2(X,\nu )}^2 \leq \frac23\left(\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3-6\right). \end{equation}
The coefficient $2/3$ is optimal, even for smooth $u$ with $\omega +\mathrm{dd}^{\mathrm{c}}u>0$. Moreover,
\[ \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3=6 \quad \Longleftrightarrow \quad u=\pi ^*v\quad \text{for some } v\in \mathrm{PSH}(\mathbb P^2,\omega _{\mathrm{FS}})\cap L^\infty (\mathbb P^2). \]
Proof.
First suppose that $u$ is smooth and $\omega +\mathrm{dd}^{\mathrm{c}}u>0$. Averaging the positive forms $\tau _ a^*(\omega +\mathrm{dd}^{\mathrm{c}}u)$ gives $\omega +\mathrm{dd}^{\mathrm{c}}Au>0$. Since $Au$ is smooth and constant on the fibers, $Au=\pi ^*v$ for a smooth function $v$ on $\mathbb P^2$. At each point, the kernel of $\partial s$ maps isomorphically onto the tangent space of the base, and $\beta $ vanishes on it. Restricting $\omega +\mathrm{dd}^{\mathrm{c}}Au=\pi ^*(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}v)+\beta $ to this kernel therefore shows that $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}v>0$.
For a smooth function $f:E\to \mathbb R$, the nonconstant Fourier characters in (23.1.11), indexed by $(p,q)\in \mathbb Z^2\setminus \{ (0,0)\} $, have eigenvalues $\pi ^2p^2+q^2/4\geq 1/4$ for $-\partial _ w\partial _{\bar w}$. Parseval's identity therefore gives
\[ \int _ E\left|f-\int _ Ef\, \mathrm{d}A_ E\right|^2\mathrm{d}A_ E \leq 4\int _ E|f_ w|^2\, \mathrm{d}A_ E. \]
Applying this inequality on every fiber, integrating against $\omega _{\mathrm{FS}}^2$ and using the mass identity (23.1.5), we obtain (23.1.39).
For bounded $u$, choose uniformly bounded smooth functions $u_ j\downarrow u$ with $\omega +\mathrm{dd}^{\mathrm{c}}u_ j>0$ by theorem 5.32.1. Their averages decrease pointwise to $Au$ by dominated convergence. The corresponding smooth base potentials $v_ j$ decrease to a bounded $\omega _{\mathrm{FS}}$-plurisubharmonic function $v$, so $Au=\pi ^*v$. Dominated convergence gives $u_ j\to u$ and $Au_ j\to Au$ in $L^2(X,\nu )$, while bounded Bedford–Taylor continuity gives convergence of the total masses. Thus (23.1.39) passes to $u$.
If the mass is $6$, the inequality gives $u=Au$ almost everywhere. Since both functions are bounded and quasi-plurisubharmonic, lemma 5.24.4 gives equality everywhere. Conversely, if $u=\pi ^*v$ with bounded $v\in \mathrm{PSH}(\mathbb P^2,\omega _{\mathrm{FS}})$, choose smooth $v_ j\downarrow v$ with $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}v_ j>0$ by theorem 5.32.1. The pullbacks decrease to $u$ and are uniformly bounded. They are strictly $\omega $-plurisubharmonic because the base forms are positive and $\beta $ is positive on the vertical line. Their vertical derivatives vanish, so (23.1.5) gives mass $6$ for each of them. Bounded Bedford–Taylor continuity gives the same mass for $u$.
It remains to prove that the constant $2/3$ is optimal. Choose a nonzero real $\rho \in C_ c^\infty (U_0)$. For $\delta \in \mathbb R$, the function
\[ u(\xi ,w)=\delta \rho (\xi )\cos (\operatorname {Im}w) \]
extends by zero to a smooth function on $X$. For sufficiently small nonzero real $\delta $, it satisfies $\omega +\mathrm{dd}^{\mathrm{c}}u>0$. Its fiber average is zero, and its only fiber frequencies are $(p,q)=(0,\pm 1)$, with eigenvalue $1/4$. Hence equality holds in the fiber inequality, and therefore in (23.1.39), with nonzero left-hand side. No smaller coefficient can hold.
$\square$
Corollary 23.1.40.09DD8DF2EF76 The Hopf metric satisfies
\[ v_-(\omega )=6,\qquad v_+(\omega )=+\infty . \]
Proof.
The lower bound $v_-(\omega )\geq 6$ follows from (23.1.39), and it is attained by constants. Theorem 23.1.1 gives $v_+(\omega )=+\infty $.
$\square$
In particular, for any sequence of bounded potentials whose masses tend to $6$, the differences from their fiber averages tend to zero in $L^2(X,\nu )$. This controls only the variation along the fibers; the averaged potentials can still be arbitrary bounded potentials on the base.
For nonpositive $u\in \mathrm{PSH}(X,\omega )$ on this threefold, proposition 13.15.9 gives finite damped measures for the weight $(1-u)^{-1}[\log (\mathrm{e}-u)]^{-a}$, $a>1$, at the threshold $r_3=1$, with uniform tails and capacity continuity. The optimal damping rate for all normalized potentials on this threefold remains open. Under unitary rotations of the last two coordinates, theorem 23.1.62 gives the exact square-root threshold.
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