{"paper":{"title":"Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.SP"],"primary_cat":"math.AP","authors_text":"Alan McIntosh, Andreas Ros\\'en, Lashi Bandara","submitted_at":"2016-03-11T14:44:35Z","abstract_excerpt":"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 f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03647","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}