Spectral flow of Callias operators, odd K-cowaist, and positive scalar curvature

TL;DR

This paper explores the spectral flow of Callias operators on complete manifolds, introduces an odd-dimensional K-cowaist, and uses these tools to establish obstructions and estimates related to positive scalar curvature.

Contribution

It introduces an intrinsic odd K-cowaist and applies spectral flow techniques to study positive scalar curvature obstructions on spin manifolds.

Findings

Infinite odd K-cowaist obstructs positive scalar curvature metrics.

Derived scalar curvature estimates extend previous results.

Established codimension formulas for spectral flow on manifolds of different dimensions.

Abstract

On a complete Riemannian manifold $M$, we study the spectral flow of a family of Callias operators. We derive a codimension zero formula when the dimension of $M$ is odd and a codimension one formula when the dimension of $M$ is even. These can be seen as analogues of Gromov--Lawson's relative index theorem and classical Callias index theorem, respectively. Secondly, we introduce an intrinsic definition of K-cowaist on odd-dimensional manifolds, making use of the odd Chern character of a smooth map from the manifold to a unitary group. It behaves just like the usual K-cowaist on even-dimensional manifolds. We then apply the notion of odd K-cowaist and the tool of spectral flow to investigate problems related to positive scalar curvature on spin manifolds. In particular, we prove infinite odd K-cowaist to be an obstruction to the existence of PSC metrics. We obtain quantitative scalar curvature estimates on complete non-compact manifolds and scalar-mean curvature estimates on compact manifolds with boundary. They extend several previous results optimally, which unfolds a major advantage of our method via spectral flow and odd K-cowaist.