A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound

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 $\mathrm{Sec}_g\leq\bar k\leq 0$ and $\mathrm{Ric}_g\geq (n-1)\underline k$, 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.