01 · The idea
In everyday language
A manifold’s topology and complex structure do not by themselves fix its distances. A metric supplies those distances and determines curvature. Calabi asked whether, within a fixed Kähler class on a compact Kähler manifold, one could choose a metric having a prescribed Ricci form representing the first Chern class. Yau proved that one can, by solving a nonlinear complex Monge–Ampère equation and obtaining the estimates needed to prevent degeneration. When the real first Chern class vanishes, the theorem supplies a Ricci-flat metric. Such a metric can still possess nonzero full curvature, so its geometry can remain rich.
Why it matters
The theorem established a fundamental bridge between complex geometry, topology and nonlinear analysis, producing Ricci-flat spaces that became important in geometry and theoretical physics.
