Proposition 14.12.1.706EB82AB6A2 On the open set of smooth potentials $u$ with $\Gamma _ u>0$, the derivative of
\[ \mathcal G(u)=\log \frac{\eta _ u^ n}{\gamma ^ n} \]
is the real elliptic operator
62F6382D2CAC
14.12.2
\begin{equation} \mathcal L_ uv= \frac{n}{n-1}\frac{\eta _ u\wedge \mathcal D_\gamma v}{\eta _ u^ n}. \end{equation}
It has no zeroth-order term, and its kernel consists of the constants. There is a unique smooth positive function $\rho _ u$, normalized by $\int _ X\rho _ u\eta _ u^ n=1$, such that the probability measure
\[ \mathrm{d}\mu _ u^\gamma =\rho _ u\eta _ u^ n \]
satisfies $\int _ X\mathcal L_ uv\, \mathrm{d}\mu _ u^\gamma =0$ for every smooth real $v$.
Proof.
For a smooth path of potentials $u(t)$, write $\eta =\eta _{u(t)}$ and $\Gamma =\Gamma _{u(t)}$. Differentiating $\eta ^{n-1}=\Gamma $ gives $(n-1)\dot\eta \wedge \eta ^{n-2}=\dot\Gamma $, where $\dot\Gamma =\mathcal D_\gamma \dot u$. Wedging with $\eta $ and dividing by $\eta ^ n$, we obtain
\[ \frac{\mathrm{d}}{\mathrm{d}t}\log \eta ^ n =\frac{n}{n-1}\frac{\eta \wedge \dot\Gamma }{\eta ^ n}, \]
which proves (14.12.2).
Only the second-order term $\mathrm{dd}^{\mathrm{c}}v\wedge \gamma ^{n-2}$ of $\mathcal D_\gamma v$ contributes to the principal symbol. For a nonzero complex covector $\zeta $ of type $(1,0)$, the form
\[ \eta _ u\wedge \mathrm i\zeta \wedge \overline\zeta \wedge \gamma ^{n-2} \]
is a positive volume form. Hence the principal symbol is positive definite, and by compactness the operator is uniformly elliptic for each fixed $u$. The remaining term is of first order, and constants are annihilated, so the strong maximum principle gives $\ker \mathcal L_ u=\mathbb R$.
The Fredholm and positivity arguments in the proof of theorem 11.4.2 apply to this real elliptic operator with first-order terms and no zeroth-order term. Its index is zero by deformation to a Laplacian, so its formal adjoint with respect to $\eta _ u^ n$ has a one-dimensional kernel spanned by a smooth strictly positive function. Normalizing its integral to one gives $\rho _ u$ and proves uniqueness.
$\square$
Pome
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