In dimensions two and three, if two positive forms $A,B$ have equal determinants, an algebraic expression quadratic in $A-B$ has a strict sign unless the two forms agree. On a surface its integral vanishes because it is exact. On a threefold the integral is a term involving $\mathrm{dd}^{\mathrm{c}}\omega $, so a sign condition on $\mathrm{dd}^{\mathrm{c}}\omega $ leads to the same conclusion. Approximation in the Sobolev norm justifies the integral identities for $W^{2,n}$ potentials.
In this section $n\in \{ 2,3\} $. Norms of coefficients of forms are taken with respect to a fixed smooth metric and $\mathrm{d}V$.
Theorem 14.23.1.AF9D036E600F Let $u,v\in W^{2,n}(X,\mathbb R)$, and assume that $A,B\geq 0$ almost everywhere and
\[ A^ n=B^ n=f\, \mathrm{d}V,\qquad f>0\quad \hbox{almost everywhere}. \]
If $n=3$, assume in addition that
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14.23.2
\begin{equation} \mathrm{dd}^{\mathrm{c}}\omega \wedge \mathrm{i}\eta \wedge \bar\eta \leq 0 \end{equation}
for every $(1,0)$-covector $\eta $; for $n=2$ no condition on $\omega $ is needed. Then $\mathrm{dd}^{\mathrm{c}}(u-v)=0$ in the sense of distributions, and $u-v$ is almost everywhere constant. For the $\omega $-plurisubharmonic representatives of $u$ and $v$, the difference is constant everywhere.
Proof.
Put
\[ \begin{gathered} s=(u+v)/2,\qquad D=\mathrm{dd}^{\mathrm{c}}w=A-B,\\ C=(A+B)/2=\omega +\mathrm{dd}^{\mathrm{c}}s. \end{gathered} \]
The coefficients of $C$ and $D$ lie in $L^ n$, so their products of top degree are integrable by Hölder's inequality. At almost every point the common determinant is positive, so $A$ and $B$ are positive definite there, and we may diagonalize them simultaneously by congruence: for a suitable coframe of $(1,0)$-covectors $\theta _1,\ldots ,\theta _ n$, write $A=\sum e_ j$, $B=\sum \lambda _ j e_ j$, where $e_ j=\mathrm{i}\theta _ j\wedge \bar\theta _ j$, $\lambda _ j>0$, and $\prod \lambda _ j=1$.
In dimension two,
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14.23.3
\begin{equation} D^2=A^2+B^2-2A\wedge B =2(2-\lambda _1-\lambda _2)e_1\wedge e_2\leq 0, \end{equation}
with equality if and only if $A=B$. In dimension three,
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14.23.4
\begin{equation} 2C\wedge D^2=A^3+B^3-A^2\wedge B-A\wedge B^2\leq 0. \end{equation}
Indeed, with $E=e_1\wedge e_2\wedge e_3$, the four terms on the right are
\[ 6E,\qquad 6E,\qquad 2\Bigl (\sum _ j\lambda _ j\Bigr )E, \qquad 2\Bigl (\sum _{i<j}\lambda _ i\lambda _ j\Bigr )E. \]
By the arithmetic–geometric mean inequality and $\prod _ j\lambda _ j=1$, each sum is at least three, and equality forces all $\lambda _ j=1$, that is, $A=B$.
Choose smooth real functions $w_ k\to w$ in $W^{2,n}$ and set $D_ k=\mathrm{dd}^{\mathrm{c}}w_ k$; such approximations are obtained by a partition of unity and convolution, and they need not preserve semipositivity. For $n=2$, Stokes' theorem gives $\int _ XD_ k^2=0$, since $D_ k^2=\mathrm{d}(\mathrm d^{\mathrm c}w_ k\wedge D_ k)$. For a fixed norm-comparison constant $C_0>0$,
\[ \| D_ k^2-D^2\| _{L^1} \leq C_0\| D_ k-D\| _{L^2}(\| D_ k\| _{L^2}+\| D\| _{L^2})\longrightarrow 0. \]
Thus $\int _ XD^2=0$, and the nonpositive integrand in (14.23.3) vanishes almost everywhere. By the equality case, $D=0$ almost everywhere.
For $n=3$, also choose smooth functions $s_ k\to s$ in $W^{2,3}$, and put $C_ k=\omega +\mathrm{dd}^{\mathrm{c}}s_ k$. We need the identity
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14.23.5
\begin{equation} \int _ X C_ k\wedge D_ k^2 =\int _ X\omega \wedge D_ k^2 =-\int _ X\mathrm{d}w_ k\wedge \mathrm d^{\mathrm c}w_ k\wedge \mathrm{dd}^{\mathrm{c}}\omega . \end{equation}
The first equality holds because $\mathrm{dd}^{\mathrm{c}}s_ k\wedge D_ k^2=\mathrm{d}(\mathrm d^{\mathrm c}s_ k\wedge D_ k^2)$. For the second equality, including its sign, put $J_ k=\mathrm{d}w_ k\wedge \mathrm d^{\mathrm c}w_ k$. Using $\mathrm d^{\mathrm c}\mathrm{d}=-\mathrm{dd}^{\mathrm{c}}$ and $(\mathrm d^{\mathrm c})^2=0$,
\[ \mathrm d^{\mathrm c}J_ k=-D_ k\wedge \mathrm d^{\mathrm c}w_ k,\qquad \mathrm{dd}^{\mathrm{c}}J_ k=-D_ k^2. \]
Stokes, first for $\mathrm{d}(\omega \wedge \mathrm d^{\mathrm c}J_ k)$ and then for $\mathrm d^{\mathrm c}(\mathrm{d}\omega \wedge J_ k)$, gives
\begin{align*} \int _ X\omega \wedge \mathrm{dd}^{\mathrm{c}}J_ k& =-\int _ X\mathrm{d}\omega \wedge \mathrm d^{\mathrm c}J_ k,\\ 0& =-\int _ X\mathrm{dd}^{\mathrm{c}}\omega \wedge J_ k-\int _ X\mathrm{d}\omega \wedge \mathrm d^{\mathrm c}J_ k. \end{align*}
Combining them yields (14.23.5).
With a fixed norm-comparison constant $C_0>0$, strong $L^3$ convergence implies convergence in $L^1$ of all products of three factors. For example,
\[ \| (C_ k-C)\wedge D_ k^2\| _1 \leq C_0\| C_ k-C\| _3\| D_ k\| _3^2\longrightarrow 0, \]
and
\[ \| C\wedge (D_ k^2-D^2)\| _1 \leq C_0\| C\| _3\| D_ k-D\| _3(\| D_ k\| _3+\| D\| _3)\longrightarrow 0. \]
Also $\mathrm{d}w_ k\wedge \mathrm d^{\mathrm c}w_ k\to \mathrm{d}w\wedge \mathrm d^{\mathrm c}w$ in $L^{3/2}$, so the right side of (14.23.5) converges. Consequently
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14.23.6
\begin{equation} \int _ XC\wedge D^2=-\int _ X\mathrm{d}w\wedge \mathrm d^{\mathrm c}w\wedge \mathrm{dd}^{\mathrm{c}}\omega \geq 0. \end{equation}
The sign follows from $\mathrm{d}w\wedge \mathrm d^{\mathrm c}w=(\mathrm{i}/(2\pi ))\partial w\wedge \bar\partial w$ and (14.23.2). The integrand on the left is nonpositive by (14.23.4), so it vanishes almost everywhere, and the equality case, which applies where the common determinant is positive, gives $D=0$ almost everywhere.
In both dimensions, therefore, $\mathrm{dd}^{\mathrm{c}}w=0$ in the sense of distributions. By proposition 5.22.2 with background zero, $w$ has a plurisubharmonic representative on $X$. It attains its maximum on the compact manifold $X$, so the maximum principle and connectedness make it constant.
For the $\omega $-plurisubharmonic representatives, lemma 5.24.3 upgrades the almost-everywhere equality $u=v+\text{constant}$ to equality everywhere. In particular, normalizing the means gives equality in the Sobolev class, and normalizing the suprema gives equality of the plurisubharmonic representatives.
$\square$
Pome
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