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

2016/03/31 by Lashi Bandara, Alan McIntosh, Andreas Rosén · 1 citation
Mathematics · #math.AP #math.DG #math.SP #msc:58J05 #msc:58J37 #msc:58J30 #msc:35J46 #msc:42B37

paper · pdf · doi:10.1007/s00208-017-1610-7

published as Math. Ann. (2018) 370: 863

arxiv created 2017/10/11 · arxiv updated 2019/07/04

Abstract

We prove that the Atiyah-Singer Dirac operator \mathrm D_{\mathrm g} in \mathrm L² depends Riesz continuously on \mathrm L^∞ perturbations of complete metrics \mathrm g on a smooth manifold. The Lipschitz bound for the map \mathrm g → \mathrm D_{\mathrm g}(1 + \mathrm D_{\mathrm 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 Calderón'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.

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• Continuity for the spectral propinquity of the Dirac operators associated with an analytic path of Riemannian metrics