Calabi conjecture

The Calabi conjecture is a significant result in differential geometry, particularly in the study of Kähler manifolds. Formulated by Eugenio Calabi in the 1950s, the conjecture addresses the existence of Kähler metrics with special properties on certain compact complex manifolds. Specifically, the conjecture states that for a given compact Kähler manifold with a vanishing first Chern class, there exists a unique Kähler metric in each Kähler class that is Ricci-flat.