Calabi conjecture

Remarks

Calabi proved uniqueness, and Yau proved existence; the result is often called the Calabi–Yau theorem. Write $\omega' = \omega + i\partial\bar\partial\varphi$ for a real function $\varphi$, by the $\partial\bar\partial$-lemma. Since the Ricci form is $-i\partial\bar\partial \log$ of the volume density, the problem becomes the complex Monge–Ampère equation $$(\omega + i\partial\bar\partial\varphi)^n = e^{F} \omega^n,$$ for a given function $F$ normalized so that $\int_M e^F \omega^n = \int_M \omega^n$. Yau solved it by the continuity method, and the key step is an a priori bound on $\sup \abs{\varphi}$.

The main application is the case $c_1(M) = 0$ in real cohomology. Take $\rho' = 0$: each Kähler class then contains a unique Ricci-flat Kähler metric. Such manifolds are the Calabi–Yau manifolds, for example K3 surfaces and quintic threefolds in $\C P^4$. The same method gives Kähler–Einstein metrics with negative constant when $c_1(M) < 0$, a result of Aubin and Yau.

Going deeper

For a student

The theorem is a statement about a fully nonlinear elliptic equation, and its proof was a landmark of geometric analysis. Its best-known consequence is the existence of Ricci-flat metrics on compact Kähler manifolds with vanishing first Chern class. These metrics have restricted holonomy in $SU(n)$ and are used in string theory and in mirror symmetry. The positive case $c_1(M) > 0$ is not covered and is the subject of the Yau–Tian–Donaldson conjecture.