01 · A life in mathematics
The person behind the work
Shing-Tung Yau was born in Shantou, Guangdong, in April 1949 and grew up in Hong Kong. He was a professor at the Institute for Advanced Study when he received the 1982 Fields Medal. His work made nonlinear partial differential equations a central tool of geometry. He proved the Calabi conjecture, establishing the existence of Kähler metrics with prescribed Ricci curvature under the required compatibility conditions. In particular, a compact Kähler manifold with vanishing real first Chern class admits a Ricci-flat Kähler metric. With Richard Schoen he also proved positive-mass results in general relativity. Ricci-flat geometry need not be flat in every curvature direction.
The contribution, explained
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.