PomeTheorem 14.2.1 (Tosatti–Weinkove).F33DA8C9DF9F For every smooth real $F$, (14.1.1) has a unique pair $(u,b)\in C^\infty (X,\mathbb R)\times \mathbb R$ with $\sup _ Xu=0$.
14.2 Scalar bounds and uniqueness5E342E12EF34
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Evaluating the equation at the extrema of the potential bounds the normalization scalar in terms of the prescribed data. Applied to the difference of two solutions, the same argument shows that their scalars agree, and ellipticity then determines the potential up to a constant. The scalar bounds will also give uniform control along the continuity path.
In dimension one, integration determines $\mathrm{e}^ b=\int _ X\omega /\int _ X\mathrm{e}^ F \mathrm{d}V$, and the remaining equation $\mathrm{dd}^{\mathrm{c}}u=\mathrm{e}^{F+b}\mathrm{d}V-\omega $ is a Poisson equation whose right-hand side has integral zero, so it has a smooth solution, unique up to a constant; positivity of $\omega _ u$ is automatic. Thus the theorem holds in that dimension, and we prove it for $n\geq 2$ at the end of section 14.6.
Proposition 14.2.2.853DC10A2ADA Every solution of (14.1.1) satisfies
\[ \inf _ X\! \left(\log \frac{\omega ^ n}{\mathrm{d}V}-F\right) \leq b\leq \sup _ X\! \left(\log \frac{\omega ^ n}{\mathrm{d}V}-F\right). \]
Proof. At a maximum point $x_+$ of $u$, the real $(1,1)$-form $\mathrm{dd}^{\mathrm{c}}u$ is nonpositive, so $0<\omega _ u\leq \omega $ there. Hence $\omega _ u^ n(x_+)\leq \omega ^ n(x_+)$, and $b\leq \log (\omega ^ n/\mathrm{d}V)(x_+)-F(x_+)$. At a minimum point the reverse argument gives the lower bound. $\square$
When $\mathrm{d}\omega =0$, integration gives the familiar explicit formula
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14.2.3
\begin{equation} b=\log \frac{\int _ X\omega ^ n}{\int _ X\mathrm{e}^ F \mathrm{d}V}. \end{equation}
The same formula holds whenever the mass-invariance criterion of theorem 11.1.7 is satisfied. In general, integrating the equation gives only
\[ \mathrm{e}^ b=\frac{\mathrm M_\omega (u)}{\int _ X\mathrm{e}^ F \mathrm{d}V}, \]
where $\mathrm M_\omega (u)=\int _ X\omega _ u^ n$ is the mass of theorem 11.1.5; the numerator depends on the unknown potential.
Proposition 14.2.4.366AA2E61158 Suppose $(u,b)$ and $(v,c)$ are smooth solutions of (14.1.1) with the same $F$ and $\mathrm{d}V$. Then $b=c$ and $u-v$ is constant.
Proof. At a maximum of $u-v$, one has $\omega _ u\leq \omega _ v$, so the equations give $\mathrm{e}^ b\leq \mathrm{e}^ c$. At a minimum one obtains $\mathrm{e}^ b\geq \mathrm{e}^ c$. Therefore $b=c$.
Put $h=u-v$ and $\eta _ t=\omega _ v+t\mathrm{dd}^{\mathrm{c}}h=(1-t)\omega _ v+t\omega _ u$ for $0\leq t\leq 1$. These forms are strictly positive. Subtracting the logarithms of the two equations and writing the difference as the integral of its derivative along the segment, we obtain
\[ 0=\int _0^1 n\, \frac{\mathrm{dd}^{\mathrm{c}}h\wedge \eta _ t^{n-1}}{\eta _ t^ n}\, \mathrm{d}t. \]
The right-hand side is a linear second-order operator applied to $h$. Its coefficients are smooth, its principal part is uniformly elliptic on the compact manifold, and it has no zeroth-order term. The strong maximum principle therefore implies that $h$ is constant. $\square$
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