Preprint

Inserted: 16 sep 2026
Last Updated: 16 sep 2026

Year: 2026

Abstract:

We prove the generalized Cartan-Hadamard conjecture, also known as the Aubin conjecture, in the small volume regime under a lower Ricci curvature bound, for any dimension $n\geq 2$. More precisely, we show that if $(M^n,g)$ is a Cartan-Hadamard manifold satisfying $Sec_g\leq\bar k\leq 0$ and $Ric_g\geq (n-1)\underline{k}$, then its isoperimetric profile is at least that of the model space of curvature $\bar k$ for small enough volumes.

We also obtain a substantial reduction of the problem for arbitrary volumes to the rigidity of the equality case.