Width, Largeness and Index Theory

TL;DR

This paper reviews recent index-theoretic advances in scalar curvature geometry, focusing on width conjectures, non-existence of positive scalar curvature metrics, and geometric bounds for manifolds, highlighting new unified insights and open problems.

Contribution

It introduces a general geometric framework linking width, largeness, and scalar curvature bounds via index theory, extending previous conjectures and results.

Findings

Unified geometric statement for width and scalar curvature bounds

Counterexamples to positive scalar curvature on certain manifolds

Open problems in index theory and geometric largeness

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.