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

Autoren: Lashi Bandara, Alan McIntosh, Andreas Rosén (2017)

We prove that the Atiyah-Singer Dirac operator Dg in L2 depends Riesz continuously on L perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map g→Dg(1+Dg2)-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.

Math. Ann.

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