Width, Largeness and Index Theory

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.

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Cited by in corpus (10)

• Scalar and mean curvature comparison via the Dirac operator
• A quantitative relative index theorem and Gromov's conjectures on positive scalar curvature
• Nonnegative scalar curvature on manifolds with at least two ends
• Quantitative K-theory, positive scalar curvature, and band width
• A proof of Gromov's cube inequality on scalar curvature
• Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension
• Spectral flow of Callias operators, odd K-cowaist, and positive scalar curvature
• Product Inequalities for ${\mathbb{T}}^{⋊}$-Stabilized Scalar Curvature
• The odd-dimensional long neck problem via spectral flow
• Macroscopic band width inequalities