We develop two new methods of constructing sequences of manifolds with positive scalar curvature that converge in the Gromov-Hausdorff and Intrinsic Flat sense to limit spaces with “pulled regions”. The examples created rigorously using these methods were announced a few years ago and have influenced the statements of some of Gromov's conjectures concerning sequences of manifolds with positive scalar curvature. Both methods extend the notion of “sewing along a curve” developed in prior work of the authors with Dodziuk to create limits that are pulled string spaces. The first method allows us to sew any compact set in a fixed initial manifold to create a limit space in which that compact set has been scrunched to a single point. The second method allows us to edit a sequence of regions or curves in a sequence of distinct manifolds.

我们开发了两种构建具有正标量曲率的流形序列的新方法，这些流形在 Gromov-Hausdorff 和内在平坦意义上会聚，以限制具有“拉动区域”的空间。使用这些方法严格创建的例子是几年前公布的，并影响了格罗莫夫关于具有正标量曲率的流形序列的一些猜想的陈述。这两种方法都扩展了作者与 Dodziuk 先前工作中开发的“沿曲线缝合”的概念，以创建拉线空间的限制。第一种方法允许我们在固定的初始流形中缝合任何紧集，以创建一个极限空间，其中该紧集被压缩成一个点。第二种方法允许我们编辑一系列不同流形中的一系列区域或曲线。