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arXiv · 2609.17093

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

Marcos Agnoletto·Márcio Fabiano Da Silva·Stefano Nardulli·Reinaldo Resende
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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 $\mathrm{Sec}_g\leq\bar k\leq 0$ 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 reduce the proof of the generalized Cartan-Hadamard conjecture to establishing rigidity in the equality case.

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Marcos Agnoletto, Márcio Fabiano Da Silva, Stefano Nardulli, Reinaldo Resende. 2026-09-21. A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound. https://arxiv.org/abs/2609.17093

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