论文标题
正标曲线 - 结构和障碍物
Positive scalar curvature -- constructions and obstructions
论文作者
论文摘要
这是对问题的当前状态的调查：“哪个封闭的连接歧管$ n \ ge 5 $允许riemannian指标，其标态曲率函数无处不在？”引言简要概述了这些结果，而本文的主体讨论了这些结果证明中使用的方法。我们提到了\ pscm s的存在的拓扑障碍的两种口味：一个是weizenböck公式的结果，用于狄拉克操作员，另一个是通过考虑稳定的最小超浮标来获得的。我们谈论\ pSCM S（手术/bordism定理）的几何结构，这表明上述问题的答案仅取决于合适的bordism群体中歧管的bordism类。通过Pontryagin-Thom结构，这可以转化为稳定的同型理论，并在某些情况下完全解决，特别是对于简单地连接的歧管或具有非常特殊的基本群体的歧管。最后一部分讨论了一些开放问题。
This is a survey of the current state of the question "Which closed connected manifolds of dimension $n\ge 5$ admit Riemannian metrics whose scalar curvature function is everywhere positive?" The introduction gives a brief overview of these results, while the body of the paper discusses the methods used in the proofs of these results. We mention the two flavors of topological obstructions to the existence of \pscm s: one is a consequence of the Weizenböck formula for the Dirac operator, the other is obtained by considering stable minimal hypersurfaces. We talk about geometric constructions of \pscm s (the surgery/bordism theorem), which shows that the answer to the question above depends only the bordism class of the manifold in a suitable bordism group. Via the Pontryagin-Thom construction this can be translated into stable homotopy theory, and solved completely in some cases, in particular for simply connected manifolds, or manifolds with very special fundamental groups. The last section discusses some open questions.
