14.1 Smooth Monge–Ampère equations92B1F482D72A

Throughout this chapter, $\omega $ is a Hermitian form on $X$ and $\mathrm{d}V$ is a smooth positive volume form. Given $F\in C^\infty (X,\mathbb R)$, we seek a smooth real potential $u$ and a scalar $b\in \mathbb R$ satisfying

Here $\omega _ u=\omega +\mathrm{dd}^{\mathrm{c}}u$, and adding a constant to $u$ does not change the equation. For a closed background, integration determines $b$ from the prescribed density. For a Hermitian background, the total mass can depend on $u$, so the potential and the scalar must be found together.

The existence proof combines weighted integration, which bounds the normalized potential, with a trace estimate for the complex Hessian, which gives uniform ellipticity. The adjoint probability measure of proposition 11.23.2 supplies the compatibility condition for the linearized equation. Including $b$ as an unknown removes this obstruction, and the estimates give the compactness needed for the continuity method. A positive exponential of the potential instead makes the equation strictly monotone and gives existence without an additional scalar.

After proving existence, we differentiate the solution with respect to the prescribed logarithmic density. The same adjoint probability measure gives the first and second variations of $b$, proving concavity and a Lipschitz bound. On the Hopf threefold of section 23.1, the fiber-energy formula measures exactly how its normalization differs from the closed-background formula.

The last part of the chapter concerns a different variation of the metric. For $n\geq 2$, we deform its $(n-1)$st power through a scalar Aeppli ansatz. If the initial metric is Gauduchon, this deformation preserves both the Gauduchon condition and its Aeppli class. The first-order torsion terms have a specific matrix structure, which makes the second-derivative estimate possible; rescaling then reduces the gradient bound to a Liouville theorem on $\mathbb C^ n$. The resulting uniqueness concerns the scalar ansatz. We finish by studying the variation of its normalization constant.