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Integration against the adjoint probability measure eliminates the derivatives of the potential from the first variation of $b$. In the second variation, the second derivative of the logarithmic determinant contributes a nonnegative square with a minus sign. This gives concavity of $b$ and identifies the equality case.
Write $b(F)$ for the scalar in (14.1.1), and use the linear normalization $\int _ Xu\, \mathrm{d}V=0$. For a smooth family $F_ t$, with $t$ in an open interval $I\subset \mathbb R$, write $(u_ t,b_ t)$ for the corresponding solutions and put $\eta _ t=\omega _{u_ t}$. By proposition 14.4.1 these depend smoothly on $t$; a dot denotes differentiation in $t$.
Theorem 14.8.1.C611B2CA264A Let $A_ t=\eta _ t^{-1}\mathrm{dd}^{\mathrm{c}}\dot u_ t$ be the endomorphism of the tangent bundle that represents the real $(1,1)$-form $\mathrm{dd}^{\mathrm{c}}\dot u_ t$ with respect to $\eta _ t$; it is Hermitian with respect to $\eta _ t$. The normalization constant satisfies
11F81D837A44
14.8.2
\begin{align} \dot b_ t& =-\int _ X\dot F_ t\, \mathrm{d}\mu _{u_ t},\\ \ddot b_ t& =-\int _ X\ddot F_ t\, \mathrm{d}\mu _{u_ t} -\int _ X\operatorname {tr}(A_ t^2)\, \mathrm{d}\mu _{u_ t}.\tag{14.8.3} \end{align}
In particular, for smooth real functions $F_0,G$ on $X$ and the affine family $F_ t=F_0+tG$, the function $b_ t$ is concave. Its second derivative vanishes at a parameter value if and only if $G$ is constant on $X$.
Proof.
Differentiating the logarithm of (14.1.1) gives
\[ L_{u_ t}\dot u_ t=\dot F_ t+\dot b_ t. \]
Integration against $\mu _{u_ t}$ proves (14.8.2).
For a smooth family of positive Hermitian matrices $H_ t$, differentiation of the inverse matrix gives
\[ \frac{\mathrm{d}^2}{\mathrm{d}t^2}\log \det H_ t =\operatorname {tr}(H_ t^{-1}\ddot H_ t) -\operatorname {tr}\bigl ((H_ t^{-1}\dot H_ t)^2\bigr ). \]
Apply this pointwise to the matrices of $\eta _ t$, using $\dot\eta _ t=\mathrm{dd}^{\mathrm{c}}\dot u_ t$ and $\ddot\eta _ t=\mathrm{dd}^{\mathrm{c}}\ddot u_ t$. The second derivative of the logarithmic equation is
\[ L_{u_ t}\ddot u_ t-\operatorname {tr}(A_ t^2) =\ddot F_ t+\ddot b_ t. \]
Integrating against $\mu _{u_ t}$ at this fixed parameter proves (14.8.3); there is no need to differentiate the measure.
The eigenvalues of the Hermitian endomorphism $A_ t$ are real, so $\operatorname {tr}(A_ t^2)\geq 0$. For affine $F_ t$ we have $\ddot F_ t=0$, so the second derivative of $b_ t$ is nonpositive. If it vanishes, strict positivity of the smooth density of $\mu _{u_ t}$ implies $A_ t=0$ everywhere. Thus $\mathrm{dd}^{\mathrm{c}}\dot u_ t=0$, and the maximum principle shows that $\dot u_ t$ is constant. The first differentiated equation then gives $G=-\dot b_ t$, a constant. Conversely, if $G$ is constant, the solution metric is independent of $t$ and $b_ t=b_0-tG$.
$\square$
Corollary 14.8.4.B9C312CB7069 For two smooth functions $F,G$, the normalization constants satisfy
\[ -\sup _ X(G-F)\leq b(G)-b(F)\leq -\inf _ X(G-F). \]
In particular, $|b(G)-b(F)|\leq \| G-F\| _\infty $. If $G-F$ is nonconstant, both inequalities in the first display are strict.
Proof.
Apply (14.8.2) along $F_ t=F+t(G-F)$ and integrate over $0\leq t\leq 1$. Each $\mu _{u_ t}$ is a probability measure with strictly positive smooth density, so the integral of a nonconstant continuous function against it lies strictly between the minimum and the maximum of that function.
$\square$
Thus $b$ is a decreasing, concave, Lipschitz functional of the logarithmic density, with $b(F+a)=b(F)-a$ for every real constant $a$. By uniform approximation, it extends uniquely to a Lipschitz functional of continuous $F$. This extension concerns the scalar functional only; the existence and regularity of corresponding potentials require further estimates.
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