AI 中文总结
该研究针对具有有限渐近纳塔维数和有限渐近秩的阿达马空间，证明了维度不低于渐近秩的整周满足线性等周填充不等式，将相关结果的指数优化至最优线性情况并推广至任意有限渐近秩。
AI 中文摘要
格罗莫夫（Gromov）的一个猜想以渐近秩重新表述，预测在阿达马（Hadamard）空间中，所有维度大于或等于其渐近秩的空间均满足线性等周填充不等式，这与低于该阈值时的欧几里得型非线性行为形成对比。我们证明了具有有限渐近纳塔（Nagata）维数和有限渐近秩的阿达马空间满足该预测的线性不等式。更确切地说，每一个维度大于或等于渐近秩的整周（integral cycle）都存在一个填充，其质量与该周的质量呈线性有界关系。我们的证明基于温格（Wenger）的次欧几里得增长定理的一种新的自改进机制，该方法将温格的渐近秩-1结果以及朗（Lang）、斯塔德勒（Stadler）和乌雷奇（Urech）的近期渐近秩-2结果从任意接近1的指数提升至最优线性指数，此外，该结果还可推广到任意有限渐近秩的情况。
英文摘要
Reformulated in terms of the asymptotic rank, a conjecture by Gromov predicts a linear isoperimetric filling inequality in all dimensions greater than or equal to the asymptotic rank of a Hadamard space, in contrast to the Euclidean-type nonlinear behavior below this threshold. We prove the predicted linear inequality for Hadamard spaces with finite asymptotic Nagata dimension and finite asymptotic rank. More precisely, every integral cycle of dimension at or above the asymptotic rank admits a filling whose mass is bounded linearly in the mass of the cycle. Our proof is based on a new self-improvement mechanism for Wenger's sub-Euclidean growth theorem. This approach upgrades the asymptotic rank-one result by Wenger and the recent asymptotic rank-two result by Lang, Stadler, and Urech from exponents arbitrarily close to one to the optimal linear exponent. Moreover, the result extends to arbitrary finite asymptotic ranks.
Comments23 pages
