Corollary 12.9.107877D1AB98D3

Proof. For a small smooth function $f$, put

If $F(f)=F(g)$, then $f-g$ is constant, so the map $f\mapsto F(f)$ is injective on functions of zero $\mathrm{d}V$-mean. By its continuity in $C^{2,\alpha }$, it maps a sufficiently small neighborhood of zero in any fixed finite-dimensional space of such smooth functions into the ball, and these spaces can have arbitrary dimension.

For the path, choose a smooth real $\psi $ of zero $\mathrm{d}V$-mean whose complex Hessian at some point $x_0$ has, in an $\omega $-unitary frame, a matrix $Q_0$ that is not a multiple of the identity. A cutoff of $|z_1|^2-|z_2|^2$ in local coordinates, minus its mean, is such a function. For small real $t$ put

These normalized data tend to $F_*$ in $C^{2,\alpha }$ as $t\to 0$, and by uniqueness in theorem 14.2.1 their solution forms are the forms $\Omega _ t$. If $F_ s=F_ t$, then $\log (\Omega _ s^ n/\Omega _ t^ n)$ is the constant $\log (V_ s/V_ t)$. The signs of the complex Hessian of $(s-t)\psi $ at a maximum and at a minimum force this constant to be zero. Integrating the derivative of $\log \det $ along the segment between $\Omega _ t$ and $\Omega _ s$ then gives a uniformly elliptic equation for $(s-t)\psi $ without zeroth-order term, and the maximum principle forces $(s-t)\psi $ to be constant. Since $\psi $ is not constant, $s=t$.

If the maximizing currents for $F_ s$ and $F_ t$ agree, (12.9.9) makes $\Omega _ s^{n-1}$ and $\Omega _ t^{n-1}$ proportional at $x_0$. Positive cofactor matrices determine positive forms up to scale, so $I+sQ_0$ and $I+tQ_0$ are proportional. Since $I$ and $Q_0$ are linearly independent, comparing coefficients gives $s=t$. $\square$