Width, largeness and index theory
Abstract
In this note, we review some recent developments related to metric aspect of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit indextheoretic 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 nonexstistence 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 ($\hat{A}$)isoenlargeable spin manifolds and related notions of width for Riemannian manifolds from an indextheoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties and width.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.13754
 Bibcode:
 2020arXiv200813754Z
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Geometric Topology;
 Mathematics  KTheory and Homology
 EPrint:
 15 pages