Calabi

Calabi is a surname of Italian origin. In mathematics it is associated with several concepts and people, most notably Eugenio Calabi, a mathematician who formulated the Calabi conjecture in complex differential geometry. The conjecture concerns the existence of Kähler metrics with prescribed Ricci curvature on compact Kähler manifolds. It was later resolved by Shing-Tung Yau, and its resolution gave rise to the notion of Calabi-Yau manifolds, which are compact Kähler manifolds with vanishing first Chern class and holonomy contained in SU(n).

Calabi-Yau manifolds have become central in algebraic geometry and theoretical physics, particularly in string theory where they are used to compactify extra dimensions in certain supersymmetric models. Beyond this, the name Calabi appears in other mathematical constructs stemming from Eugenio Calabi, such as the Calabi flow, a geometric evolution equation for Kähler metrics, and the Calabi invariant (Calabi homomorphism) in symplectic geometry, which assigns a real number to certain Hamiltonian diffeomorphisms.

In summary, Calabi denotes both a family of mathematical ideas linked to differential and symplectic geometry and a surname associated with the mathematician who contributed foundational problems later developed into key concepts in geometry and physics.