The Calabi conjecture is a fundamental idea in differential geometry, proposed by mathematician Eugenio Calabi in the 1950s. It suggests that under certain conditions, a compact Kähler manifold can be equipped with a unique Kähler metric that has constant Ricci curvature. This conjecture connects complex geometry with the study of Einstein metrics. In 2000, the conjecture was proven by Shing-Tung Yau, who showed that such metrics exist and are unique. This breakthrough has significant implications in various fields, including string theory and mathematical physics, as it helps in understanding the geometry of complex spaces.