PomeLemma 14.18.1.1ECE72025D8E The solution $(u,b_\gamma (F))$, normalized by $\int _ Xu\, \mathrm{d}V_0=0$, depends smoothly on the parameters of every smooth finite-dimensional family of functions $F$.
14.18 Normalization for the Gauduchon equation9C1043361EBF
9C1043361EBF
By theorem 14.10.1, every smooth real function $F$ determines a scalar $b_\gamma (F)$, the constant $b$ of that theorem. As for the Monge–Ampère equation, its variation is governed by the adjoint probability measure $\mu _ u^\gamma $ of proposition 14.12.1 and by the concavity of the logarithmic determinant. To differentiate solutions, we fix a smooth positive volume form $\mathrm{d}V_0$ and use the normalization $\int _ Xu\, \mathrm{d}V_0=0$ in place of $\sup _ Xu=0$.
Proof. The map $\Gamma \mapsto \eta $ with $\eta ^{n-1}=\Gamma $ is smooth on positive forms, since in a frame it is the inverse of the smooth map $\eta \mapsto \eta ^{n-1}$, whose differential is invertible. The openness argument in the proof of theorem 14.10.1, with the $\mathrm{d}V_0$-mean in place of the $\gamma ^ n$-mean, gives a bounded inverse for the derivative with respect to the potential and the scalar between the relevant Hölder spaces. The implicit function theorem gives local smooth solution maps, and uniqueness shows that they agree on overlaps. Applying this for every order of Hölder regularity gives the assertion for smooth data. $\square$
For a smooth family of prescribed functions $F_ t\in C^\infty (X,\mathbb R)$, with $t$ in an open interval of $\mathbb R$, let $u_ t$ be the corresponding solutions with $\int _ Xu_ t\, \mathrm{d}V_0=0$, and set $Q_ t=Q_{u_ t}$ as in (14.13.1). The expression
\[ \operatorname {tr}\bigl ((Q_ t^{-1}\dot Q_ t)^2\bigr ) \]
is independent of the chosen frame and nonnegative, because $Q_ t^{-1}\dot Q_ t$ is similar to the Hermitian matrix $Q_ t^{-1/2}\dot Q_ t Q_ t^{-1/2}$.
Theorem 14.18.2.F90961841CA0 For a smooth family $F_ t$ and its solutions,
\begin{align*} \dot b_\gamma (F_ t)& =-\int _ X\dot F_ t\, \mathrm{d}\mu _{u_ t}^\gamma ,\\ \ddot b_\gamma (F_ t)& =-\int _ X\ddot F_ t\, \mathrm{d}\mu _{u_ t}^\gamma -\frac1{n-1}\int _ X \operatorname {tr}\bigl ((Q_ t^{-1}\dot Q_ t)^2\bigr ) \, \mathrm{d}\mu _{u_ t}^\gamma . \end{align*}
In particular, $b_\gamma $ is concave, and along $F_ t=F_0+tG$ its second derivative is strictly negative unless $G$ is constant. Moreover, for smooth real functions $F,G$ on $X$ and every real constant $a$,
\[ b_\gamma (F+a)=b_\gamma (F)-a, \qquad |b_\gamma (G)-b_\gamma (F)|\leq \| G-F\| _\infty . \]
Proof. By (14.13.1),
\[ \mathcal G(u)=\frac1{n-1}\log \det _\gamma Q_ u. \]
Differentiating $\mathcal G(u_ t)=F_ t+b_\gamma (F_ t)$ once gives $\mathcal L_{u_ t}\dot u_ t=\dot F_ t+\dot b_\gamma (F_ t)$, and integration against the adjoint probability measure gives the first formula.
Because $\Gamma _ u$ is affine in $u$, its second derivative along the family is $\mathcal D_\gamma \ddot u_ t$. The formula for the second derivative of $\log \det $ used in the proof of theorem 14.8.1 therefore gives
\[ \mathcal L_{u_ t}\ddot u_ t -\frac1{n-1}\operatorname {tr}\bigl ((Q_ t^{-1}\dot Q_ t)^2\bigr ) =\ddot F_ t+\ddot b_\gamma (F_ t). \]
Integrating against $\mu _{u_ t}^\gamma $ at this fixed parameter gives the second formula; no derivative of the measure is needed.
For affine $F_ t$, we have $\ddot F_ t=0$, and positivity of the density of $\mu _{u_ t}^\gamma $ proves concavity. If the second derivative vanishes, then $\dot Q_ t=0$, so $\dot\Gamma _{u_ t}=0$ and $\mathcal L_{u_ t}\dot u_ t=0$. By proposition 14.12.1, $\dot u_ t$ is constant, and the first differentiated equation forces $G$ to be constant. Conversely, adding a constant to $F$ changes only $b$, by the opposite constant. Finally, integrate the first derivative along the segment from $F$ to $G$: since $\mu _{u_ t}^\gamma $ is a probability measure, the derivative lies between $-\sup _ X(G-F)$ and $-\inf _ X(G-F)$, which proves the Lipschitz bound. $\square$
The Lipschitz bound extends $b_\gamma $ uniquely to continuous functions $F$. Constructing corresponding weak solutions would require, in addition, a volume measure for the $(n-1)$st roots of singular forms in the Aeppli ansatz, compactness of their potentials, and stability of the adjoint equation.
On a surface with Gauduchon background form $\gamma _0$, conservation of the total mass makes the scalar and the adjoint probability measure explicit.
Example 14.18.3.8AC1A3B0F2AF If $n=2$ and $\gamma _0$ is Gauduchon, then $\Gamma _ u=\gamma _0+\mathrm{dd}^{\mathrm{c}}u=\eta _ u$, and $\eta _ u$ is also Gauduchon. The mass identity of theorem 11.1.7 gives $\int _ X\eta _ u^2=\int _ X\gamma _0^2$. Therefore
\[ b_\gamma (F)=\log \int _ X\gamma _0^2- \log \int _ X\mathrm{e}^ F\gamma ^2, \qquad \mathrm{d}\mu _ u^\gamma =\frac{\mathrm{e}^ F\gamma ^2}{\int _ X\mathrm{e}^ F\gamma ^2}. \]
The adjoint equation of proposition 14.12.3 gives the same measure, because $\mathrm{dd}^{\mathrm{c}}\eta _ u=0$ and hence $\rho _ u$ is constant. Along $F+tG$, the second derivative of $b_\gamma $ is minus the variance of $G$ with respect to this probability measure.
In section 14.19 we return to the potential equation for $\omega +\mathrm{dd}^{\mathrm{c}}u$. Smooth approximation will extend its solvability to nonnegative, nonzero $L^ p$ densities, $p>1$; control of the normalization constant and uniform convergence of the potentials will replace the derivative estimates used here.
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