标题： 一个具有非负标量曲率的极端极限 显示英文标题

标题： An Extreme Limit with Nonnegative Scalar Curvature

摘要： 2014年，格罗莫夫模糊地猜想，具有非负标量曲率的流形序列应该有一个子序列在某种弱意义下收敛到一个具有广义非负标量曲率概念的极限空间。 该猜想在IAS新兴主题会议上被明确：要求序列是三维的，并且直径和体积有统一的上界，以及MinA有一个正的统一下界，其中MinA是流形中闭合极小曲面的最小面积。 在这里，我们提出了一种通过标准球面的扭曲乘积流形，它们在极点上有圆纤维，其长度发散到无穷大，满足这个IAS猜想的假设。 我们证明这个序列在$W^{1,p}$的意义下对于$p<2$收敛到一个极端的极限空间，该空间在Lee-LeFloch定义的分布意义下具有非负标量曲率，并且总分布标量曲率收敛。 本文只需要光滑黎曼几何、光滑极小曲面和Sobolev空间方面的专业知识。 在第二篇论文中，需要度量几何方面的专业知识，前两位作者证明了我们的序列在内在平坦和Gromov-Hausdorff意义下收敛到这个极端的极限空间，并研究了它的几何性质。 显示英文摘要

摘要： In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. Here we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the $W^{1,p}$ sense for $p<2$ to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. This paper only requires expertise in smooth Riemannian Geometry, smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring expertise in metric geometry, the first two authors prove intrinsic flat and Gromov-Hausdorff convergence of our sequence to this extreme limit space and investigate its geometric properties.