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Calabi transformed the Calabi conjecture into a non–linear partial differential equation of complex Monge–Ampere type, and showed that this equation has at most one solution, thus establishing the uniqueness of the required Kähler metric.
A Calabi–Yau manifold can be defined as a compact Ricci-flat Kähler manifold or equivalently one whose first Chern class vanishes.
On the other hand, Mirror symmetry allows for the mathematical similarity between two distinct Calabi-Yau manifolds.
Homological mirror symmetry, a mathematical conjecture about Calabi-Yau manifolds made by Maxim Kontsevich