Calabi conjecture / complex Monge–Ampère existence

Define ${\mu }_t=(1-t){\omega }_0^n+t\mu$ for $0\le t\le 1$. This is a smooth path of positive volume forms because it is a convex combination of positive top-degree forms, and ${\int }_X{\mu }_t=(1-t)V+tV=V$. Set $T$ equal to the set of $t$ for which there is a smooth real normalized ${\varphi }_t$ with ${\omega }_{{\varphi }_t}:={\omega }_0+i\mathrm{\partial }\bar{\mathrm{\partial }}{\varphi }_t>0$ and ${\omega }_{{\varphi }_t}^n={\mu }_t$.

At $t=0$, ${\varphi }_0=0$ is normalized and solves the equation, so $0\in T$. For every positive potential define the density residual $\mathcal{F}(t,\varphi )={\omega }_{\varphi }^n\mathrm{/}{\omega }_0^n-{\mu }_t\mathrm{/}{\omega }_0^n$. Its integral against ${\omega }_0^n$ is zero: the volume forms in the same Kähler class have equal total volume by expanding ${\omega }_{\varphi }^n-{\omega }_0^n$ as an exact form and applying [F4], and ${\int }_X{\mu }_t=V$.

By [F5], any sequence $t_j\in T$ converging to $t_{\mathrm{\infty }}$ has corresponding normalized solutions uniformly bounded in every $C^m$. Apply [F3] to obtain a smooth normalized limit with positive Kähler form solving the equation at $t_{\mathrm{\infty }}$. Hence $t_{\mathrm{\infty }}\in T$ and $T$ is closed in $[0,1]$.

At a solution, differentiating the determinant gives $D_{\varphi }\mathcal{F}[\psi ]=({\mathrm{tr}}_{{\omega }_{\varphi }}(i\mathrm{\partial }\bar{\mathrm{\partial }}\psi ))({\omega }_{\varphi }^n\mathrm{/}{\omega }_0^n)$. The trace operator is uniformly elliptic at this fixed positive metric. Differentiating the zero-integral identity from step 2.1 shows the derivative has zero ${\omega }_0^n$-mean. If it vanishes, then the trace vanishes; Stokes [F4] gives ${\int }_{X}i\mathrm{\partial }\psi \wedge \bar{\mathrm{\partial }}\psi \wedge {\omega }_{\varphi }^{n-1}=-{\int }_X\psi i\mathrm{\partial }\bar{\mathrm{\partial }}\psi \wedge {\omega }_{\varphi }^{n-1}=0$, so $\mathrm{\partial }\psi =0$ and $\psi$ is constant. Restricting to zero ${\omega }_0^n$-mean removes that kernel. The exact weighted Hölder isomorphism is supplied by [F6]: solving for a target $f$ of zero reference mean requires ${\mathrm{\Delta }}_{{\omega }_{\varphi }}\psi =f\mathrm{/}\rho$, whose compatibility is ${\int }_X(f\mathrm{/}\rho ){\omega }_{\varphi }^n={\int }_{X}f{\omega }_0^n=0$. Its constant is then fixed by the reference normalization. Thus the fixed domain and target here match the actual inverse, rather than imposing the wrong mean on the unweighted Laplacian.

The map $(t,\varphi )\mapsto \mathcal{F}(t,\varphi )$ is smooth in the Hölder spaces on the open positive cone. By [F6] and step 3.1, its derivative in the normalized potential variable is a bounded isomorphism between the fixed-reference mean-zero spaces. The Banach implicit-function argument of [F6], using $k=1$, gives positive normalized $C^{3,\alpha }$ solutions for all sufficiently close density parameters; the difference-quotient Schauder bootstrap proved there makes them smooth. The density-neighbourhood formulation works also at the endpoints, with parameters restricted to $[0,1]$. Hence $T$ is relatively open.

The interval $[0,1]$ is connected; since $T$ is nonempty by step 2.1, open by step 4.1, and closed by step 2.2, $T=[0,1]$. In particular a normalized solution exists at $t=1$.

Suppose $\varphi$ and $\psi$ are two normalized solutions at $t=1$. Put $u=\varphi -\psi$ and $S={\sum }_{j=0}^{n-1}{\omega }_{\varphi }^j\wedge {\omega }_{\psi }^{n-1-j}$, a closed positive $(n-1,n-1)$-form. The algebraic identity for the difference of powers gives $0={\omega }_{\varphi }^n-{\omega }_{\psi }^n=i\mathrm{\partial }\bar{\mathrm{\partial }}u\wedge S$. Stokes [F4] applied to $ui\bar{\mathrm{\partial }}u\wedge S$ gives $0={\int }_{X}ui\mathrm{\partial }\bar{\mathrm{\partial }}u\wedge S=-{\int }_{X}i\mathrm{\partial }u\wedge \bar{\mathrm{\partial }}u\wedge S$. Positivity of $S$ forces $\mathrm{\partial }u=0$, so $u$ is constant; its zero mean forces $u=0$. Thus uniqueness is proved independently of the a priori estimates.