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 $n\ge 2$. More precisely, we show that if $(M^n,g)$ is a Cartan-Hadamard manifold satisfying ${\mathrm{S}\mathrm{e}\mathrm{c}}_g\le \bar{k}\le 0$ and ${\mathrm{R}\mathrm{i}\mathrm{c}}_g\ge (n-1)\underset{‾}{k}$, 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.