Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric

TL;DR

This paper demonstrates that the Atiyah-Singer Dirac operator varies continuously with respect to metric perturbations on a manifold, using harmonic analysis techniques and establishing bounds based on curvature and injectivity radius.

Contribution

It establishes Riesz continuity of the Dirac operator under metric perturbations and extends results to general functions of Dirac-type operators.

Findings

Riesz continuity of Dirac operators under metric changes

Bounds depend on Ricci curvature and injectivity radius

Perturbation results for functions of Dirac-type operators

Abstract

We prove that the Atiyah-Singer Dirac operator ${\mathrm D}_{\mathrm g}$ in ${\mathrm L}^2$ depends Riesz continuously on ${\mathrm L}^{\infty}$ perturbations of complete metrics ${\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\mathrm g} \to {\mathrm D}_{\mathrm g}(1 + {\mathrm D}_{\mathrm g}^2)^{-\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.