His doctoral dissertation, titled "Isometric complex analytic imbedding of Kähler manifolds", was done under the supervision of Salomon Bochner.

In the comments on his collected works in 2021, Calabi cited his article "Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens" as that which he was "most proud of".

[13] At the 1954 International Congress of Mathematicians, Calabi announced a theorem on how the Ricci curvature of a Kähler metric could be prescribed.

[14] A well-known construction of Calabi's puts complete Kähler metrics on the total spaces of hermitian vector bundles whose curvature is bounded below.

[C58a] The Riemannian distance function is generally not differentiable everywhere, which poses a difficulty in formulating a global version of the theorem.

Calabi made use of a generalized notion of differential inequalities, predating the later viscosity solutions introduced by Michael Crandall and Pierre-Louis Lions.

In analogous work, Calabi had earlier considered the convex solutions of the Monge–Ampère equation which are defined on all of Euclidean space and with 'right-hand side' equal to one.

Konrad Jörgens had earlier studied this problem for functions of two variables, proving that any solution is a quadratic polynomial.

[19] Later, Calabi considered the problem of affine hyperspheres, first characterizing such surfaces as those for which the Legendre transform solves a certain Monge–Ampère equation.

By adapting his earlier methods in extending Jörgens' theorem, Calabi was able to classify the complete affine elliptic hyperspheres.

[21][22] Inspired by recent work of Kunihiko Kodaira, Calabi and Edoardo Vesentini considered the infinitesimal rigidity of compact holomorphic quotients of Cartan domains.

[CV60] Making use of the Bochner technique and Kodaira's developments of sheaf cohomology, they proved the rigidity of higher-dimensional cases.

In his PhD thesis, Calabi had previously considered the special case of holomorphic isometric embeddings into complex-geometric space forms.

[C53] A striking result of his shows that such embeddings are completely determined by the intrinsic geometry and the curvature of the space form in question.