Along with Yau, he also showed that Kähler manifolds with negative first Chern classes always admit Kähler–Einstein metrics, a result closely related to the Calabi conjecture.

[2] Independently, Shing-Tung Yau proved the more powerful Calabi conjecture, which concerns the general problem of prescribing the Ricci curvature of a Kähler metric, via non-variational methods.

He established Riemannian formulations of many classical results for Sobolev spaces, such as the equivalence of various definitions, the density of various subclasses of functions, and the standard embedding theorems.

Along with similar results for the Moser–Trudinger inequality, Aubin later proved improvements of the optimal constants when the functions are assumed to satisfy certain orthogonality constraints.

Following prior work of Neil Trudinger, Aubin was able to resolve the problem in high dimensions under the condition that the Weyl curvature is nonzero at some point.