Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 127, 15 pages      arXiv:2008.13754      https://doi.org/10.3842/SIGMA.2020.127
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Width, Largeness and Index Theory

Rudolf Zeidler
Mathematical Institute, University of Münster, Einsteinstr. 62, 48149 Münster, Germany

Received September 01, 2020, in final form November 26, 2020; Published online December 02, 2020; Typos corrected August 14, 2024

Abstract
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.

Key words: scalar curvature; comparison geometry; index theory; Dirac operator; Callias-type operator; enlargeability; largeness properties.

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