Width, Largeness and Index Theory

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2020
• Main Author: Zeidler, Rudolf
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2020
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211092
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Width, Largeness and Index Theory. Rudolf Zeidler. SIGMA 16 (2020), 127, 15 pages

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 × [−1, 1], and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on 𝑀 × 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 × ℝ if the scalar curvature is positive in some neighborhood. We study (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.