15.31.2 The logarithmic potential and its measure trace06937C5027C2

Proof. The Poincaré–Lelong formula gives $\mathrm{dd}^{\mathrm{c}}_ xG(x,y)=\delta _ y-\kappa $. After a rotation sending $y$ to $[1:0]$, the variable $t=d_{\mathrm{ch}}(x,y)^2$ is uniformly distributed on $[0,1]$ for the measure $\kappa $. Thus

The continuous potentials $\int \max (G(x,y),-L)\, \mathrm d\mu (y)$ are $\kappa $-plurisubharmonic by lemma 5.24.1 and averaging, and decrease to $Q_\mu $. Tonelli's theorem gives its integrability and mean zero; the decreasing-limit property gives its canonical values. Distributional Fubini proves the curvature identity.

In degree one the extended measure of $p$ is

The pole set of $Q_\mu $ is polar, so non-polarity gives its asserted measure. For any $p$, distributional Stokes gives the full current $\kappa +\mathrm{dd}^{\mathrm{c}}p$ mass one, and its extended measure has mass at most one. Domination of the probability $\mu $ therefore forces both measures to equal $\mu $. The distribution $p-Q_\mu $ is harmonic on the compact connected curve, hence constant, and canonical equality gives the assertion everywhere. $\square$

Proof. Continuity gives the uniform kernel bound

To prove non-polarity, take a nontrivial subharmonic function on a coordinate disk. Cut off its positive Hessian measure on a slightly larger disk. The logarithmic potential of this finite measure has the same Hessian on the smaller disk, so their difference is harmonic there. This is the local Riesz decomposition: it also follows directly from the distributional fundamental solution and harmonic regularity. On compact subdisks the kernel $-\log |z-a|^2$ differs from $K(z,a)$ by a bounded smooth term. Symmetry of $K$, Tonelli's theorem and (15.31.6) make the logarithmic potential integrable against $\mu $; its harmonic remainder is bounded. The pole set consequently has $\mu $-mass zero. A countable disk cover proves the claim for polar Borel sets.

The group $\mathrm{SU}(2)$ acts on $Y$ by Fubini–Study isometries. Average with smooth nonnegative conjugation-invariant kernels of mass one supported near the identity. This gives smooth $Q_\epsilon $, probability measures $\mu _\epsilon $, and

Smooth Stokes gives $E(Q_\epsilon ,Q_\epsilon )=-\int Q_\epsilon \, \mathrm d\mu _\epsilon \leq \| Q\| _\infty $. Poincaré's inequality and weak Hilbert compactness show that $Q\in H^1$. Its distributional curvature identity gives (15.31.5) first for smooth $h$. Apply this identity to $\sqrt{h^2+\epsilon ^2}$, whose gradient has magnitude at most $|\mathrm dh|$. Letting $\epsilon \downarrow 0$ yields

Completion defines the trace and extends the identity.

These group averages converge strongly in $H^1$: rotations act continuously and isometrically on that space, and the averaging kernels approach the identity. They also converge pointwise to every canonical quasi-subharmonic function. Indeed their induced averages about a point are radial by conjugation invariance. In a coordinate centered at that point they are positive averages of circular means; after a smooth quadratic correction, the submean inequality and upper semicontinuity give the required limit. The correction error tends to zero, and upper semicontinuity gives the limit also at a pole. For a $\kappa $-plurisubharmonic $H^1$ function, strong convergence and the trace bound give $L^1(\mu )$ convergence, while a subsequence converges $\mu $-almost everywhere to the trace. Non-polarity makes the canonical value finite there and identifies the two. Finally, the Sobolev chain rule and smooth approximation show that Lipschitz compositions are continuous in $H^1$; the corresponding $L^1(\mu )$ limits prove the composition assertion. $\square$