论文标题
宽度，宽敞和索引理论
Width, Largeness and Index Theory
论文作者
论文摘要
在本说明中，我们回顾了一些与标量曲率的度量方面相关的最新发展，从狄拉克运营商的索引理论的角度来看。 In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands $M \times [-1,1]$, and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on $M \times {\mathbb R}$.我们表明，它们的基础上有一个更通用的几何陈述，这意味着如果标量曲率在某些邻域中为正面，则在$ m \ times {\ times {\ times {\ times {\ times {\ mathbb r} $上的标量曲率的最小值。我们从索引理论的角度研究（$ \ hat {a} $ - ）ISO-ENLARARGANGE-ENARMANGE-ENARGANGE SPIN歧管和相关宽度的宽度概念。最后，我们列出了索引理论，宽敞属性和宽度之间的相互作用中产生的一些开放问题。
In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands $M \times [-1,1]$, and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on $M \times {\mathbb R}$. We show that there is a more general geometric statement underlying both of them implying a quantitative negative upper bound on the infimum of the scalar curvature of a complete metric on $M \times {\mathbb R}$ if the scalar curvature is positive in some neighborhood. We study ($\hat{A}$-)iso-enlargeable spin manifolds and related notions of width for Riemannian manifolds from an index-theoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties and width.
