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

Journal article, 2018

We prove that the Atiyah-Singer Dirac operator D-g in L-2 depends Riesz continuously L-infinity on perturbations of complete metrics on a smooth manifold. The Lipschitz bound for the map g -> D-g(1 + D-g(2))(-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 Caldern'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.