Yau's proof of the Calabi conjecture

Yau proved the Calabi conjecture by showing that, on a compact complex manifold with a given volume form, there exists a unique Hermitian metric with a specified curvature form—specifically, a Kähler metric with a prescribed Ricci form. He did this by solving a nonlinear partial differential equation called the complex Monge-Ampère equation, using then-innovative techniques in analysis. His work established that such metrics always exist and are unique, leading to the existence of Calabi-Yau manifolds, which have exceptional geometric and physical significance, especially in string theory.