小体积下 Ricci 曲率下界情形的 Cartan-Hadamard 猜想证明
A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound
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中文总结 AI 辅助
本文在 Ricci 曲率下界条件下证明了小体积情形的广义 Cartan-Hadamard 猜想，并给出任意体积情形向等式刚性归约的关键简化。
中文摘要 AI 辅助
我们在 Ricci 曲率下界条件下，对任意维数 $n\geq 2$ 证明了小体积情形的广义 Cartan-Hadamard 猜想（亦称 Aubin 猜想）。更精确地，我们证明：若 $(M^n,g)$ 是满足 $\mathrm{Sec}_g\leq\bar k\leq 0$ 与 $\mathrm{Ric}_g\geq (n-1)\underline k$ 的 Cartan-Hadamard 流形，则当体积足够小时，其等周轮廓至少不低于曲率为 $\bar k$ 的模型空间。我们还获得了将任意体积情形的问题大幅归约为等式情形的刚性。
英文摘要
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.
发表机构
- Federal University of ABC（ABC联邦大学）
- Florida State University（佛罗里达州立大学）
机构由 AI 辅助整理，请以论文原文为准。
