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

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 $n\ge 2$. More precisely, we show that if $(M^n,g)$ is a Cartan-Hadamard manifold satisfying ${\mathrm{Sec}}_g\le \bar{k}\le 0$ and ${\mathrm{Ric}}_g\ge (n-1)\underline{k}$, then its isoperimetric profile is at least that of the model space of curvature $\bar{k}$ for small enough volumes. We also reduce the proof of the generalized Cartan-Hadamard conjecture to establishing rigidity in the equality case.

arXiv commentWe weakened Assumption 1 and added explanations for a few inequalities and steps