AI 中文总结
本文证明度量空间上配对群胚近似作用可决定Lipschitz-薄群胚作用，肯定解决Curry-Manchon缝合猜想，构造基于基点和平乐，并证明粗糙积分下降至薄群胚。
AI 中文摘要
度量空间的配对群胚的一个近似作用决定了其Lipschitz-薄群胚在完备扩展度量纤维上的一个作用，该作用由其局部缝合估计唯一刻画。我们在超线性三点缺陷估计和有限乘积的Lipschitz界条件下证明了这一断言。由此，我们获得了Curry和Manchon缝合猜想[2]的一个肯定证明，并对其估计进行了必要的路径缩放修正。进一步限制到迷向部分，构造了其注记4.15中预期的基于基点和平乐，作为基于基点的Lipschitz环路模去薄等价的一个表示。矩形比较估计给出了完全的相对Lipschitz-同伦下降，特别是在其强编织假设下以及在任意总阶大于二的三点估计下。我们证明了相应的平坦性和基点协变性陈述。一个面积模型表明该阶阈值是精确的。我们还将指数估计与一般编织假设分离开来，并展示了一个紧致圆盘度量，在该度量下强编织并不蕴含在仅连续同伦下的不变性。作为应用，对于度量空间上的受控场，我们证明了经典粗糙积分下降到薄群胚，并允许与一个公共粗糙控制器进行替换。
英文摘要
An approximate action of the pair groupoid of a metric space determines an action of its Lipschitz-thin groupoid on complete extended metric fibers, uniquely characterized by its local sewing estimate. We prove this assertion under a superlinear three-point defect estimate and a Lipschitz bound for finite products. As a result, we obtain a positive proof of Curry and Manchon's sewing conjecture [2], with the necessary path-scaling correction to its estimate. Further restriction to isotropy constructs the based holonomy anticipated in their Remark 4.15 as a representation of based Lipschitz loops modulo thin equivalence. Rectangular comparison estimates give full relative Lipschitz-homotopy descent, in particular under their strong knitting hypothesis and under any three point estimate of total order greater than two. We prove the corresponding flatness and basepoint-covariance statements. An area model shows that the order threshold is sharp. We also separate the exponential estimate from the general knitting assumptions and exhibit a compact disk metric for which strong knitting does not imply invariance under merely continuous homotopy. As an application, for controlled fields on a metric space, we prove that the classical rough integral descends to the thin groupoid and admits substitution with a common rough controller.
